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Lacunes chargées, étude dans des nano-agrégats desiliciumArpan Deb

To cite this version:Arpan Deb. Lacunes chargées, étude dans des nano-agrégats de silicium. Autre [cond-mat.other].Université de Grenoble, 2012. Français. NNT : 2012GRENY027. tel-00744666

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To my mother Mrs. Chhabi Deb.

I do not put my faith in any new institutions, but in the individuals all overthe world who think clearly, feel nobly and act rightly. They are the

channels of moral truth. ... Rabindranath Tagore

1

Contents

1 Introduction 61.1 General Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Types of defects . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Point defects . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Linear defects . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Planar defects . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 A brief history of defects in bulk Si . . . . . . . . . . . . . . . 121.4 The Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Some Theoretical Concepts : DFT and its components 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Density functional theory (DFT) (a brief overview) . . . . . . 17

2.2.1 Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . 182.2.2 The Kohn-Sham (KS) equations . . . . . . . . . . . . . 20

2.3 Different Basis Set Approaches . . . . . . . . . . . . . . . . . . 242.3.1 Atomic Basis Sets . . . . . . . . . . . . . . . . . . . . . 242.3.2 Plane Wave Basis Sets . . . . . . . . . . . . . . . . . . 262.3.3 Wavelet Basis Sets . . . . . . . . . . . . . . . . . . . . 28

2.4 Approximations to the Exchange-Correlation Potential . . . . 302.4.1 The Local Density Approximation (LDA) . . . . . . . . 302.4.2 Overview of the performance of LDA . . . . . . . . . . 312.4.3 Generalized Gradient Approximations (GGA) . . . . . 332.4.4 Meta-GGA . . . . . . . . . . . . . . . . . . . . . . . . 352.4.5 Hybrid Schemes : Combination of Hartree-Fock and

DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5.1 Normconserving Pseudopotentials . . . . . . . . . . . . 392.5.2 Fully Nonlocal Pseudopotentials . . . . . . . . . . . . . 412.5.3 Vanderbilt Ultrasoft Pseudopotentials . . . . . . . . . . 42

2

3 State of the Art: Experiments and Theory 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Experimental Methods To Study Point Defects . . . . . . . . . 463.3 Theoretical Methods To Study Point Defects . . . . . . . . . . 50

3.3.1 Treatment of Makov and Payne . . . . . . . . . . . . . 503.3.2 Treatment of Schultz . . . . . . . . . . . . . . . . . . . 553.3.3 Treatment of Freysoldt, Neugebauer and Van de Walle 593.3.4 Treatment of Dabo et.al. . . . . . . . . . . . . . . . . . 61

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4 Hartwigsen-Goedeker-Hutter(HGH) pseudopotentials with NonLinear Core Correction (NLCC) 694.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.1 Non Linear Core Correction . . . . . . . . . . . . . . . 714.3 Tests for the pseudopotentials . . . . . . . . . . . . . . . . . . 77

4.3.1 Hund’s rule . . . . . . . . . . . . . . . . . . . . . . . . 784.3.2 Calculational parameters . . . . . . . . . . . . . . . . . 794.3.3 Atomization energy of molecules using PBE . . . . . . 794.3.4 Atomization energy of molecules using PBE with NLCC 864.3.5 Similar effort with Linear Spin Density Approxima-

tion(LSDA) . . . . . . . . . . . . . . . . . . . . . . . . 944.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 Charged Defects in Silicon Nanoclusters 985.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2 Comparison of formulae (PBC vs. FBC) . . . . . . . . . . . . 102

5.2.1 PBC formalism to calculate the formation energy . . . 1025.2.2 FBC formalism to calculate the formation energy . . . 104

5.3 Density analysis of the defect systems . . . . . . . . . . . . . . 1055.3.1 Uncharged clusters . . . . . . . . . . . . . . . . . . . . 1055.3.2 Charged clusters . . . . . . . . . . . . . . . . . . . . . 108

5.4 Comparison of results obtained . . . . . . . . . . . . . . . . . 1175.5 Electrostatic model . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 1195.5.2 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5.3 Modification . . . . . . . . . . . . . . . . . . . . . . . . 1215.5.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.6 Geometry of defects in nanoclusters . . . . . . . . . . . . . . . 1225.6.1 Jahn-Teller Distorsion (JT) . . . . . . . . . . . . . . . 1225.6.2 Jahn-Teller Distorsion in the uncharged nanoclusters . 124

3

5.6.3 Jahn-Teller Distorsion in the charged nanoclusters . . . 1275.7 A simple force model to simulate the JT distorsion . . . . . . 1295.8 The combination of the electrostatic and simple force model . 1305.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.10 Future perspectives . . . . . . . . . . . . . . . . . . . . . . . . 132

5.10.1 Different other defects . . . . . . . . . . . . . . . . . . 1325.10.2 Migration energies . . . . . . . . . . . . . . . . . . . . 1335.10.3 Homothetic clusters . . . . . . . . . . . . . . . . . . . . 134

6 Final words and Perspectives 136

4

Chapitre 1

Ce chapitre traite des elements d’introduction a cette these. J’y in-

troduits la description des divers types de defauts presents dans un solide.

Certains resultats et leur importance sont aussi mentionnes. Une rapide com-

paraison avec les methodes existantes pour etudier les defauts est presente.

Les defauts decrits sont les defauts ponctuels, les defauts plans, les disloca-

tions, avec ou sans charge. Une rapide histoire de l’etude des defauts est

aussi mentionnee pour completer l’etat de l’art du domaine. Enfin, certaines

experiences portant sur les defauts sont rappelees.

5

Chapter 1

Introduction

1.1 General Ideas

Silicon (Si), the most common metalloid and the eight most common ele-

ment in the universe by mass is one among the most important material for

industrial use. It is the principal component of most semiconductor devices,

most importantly integrated circuits or microchips. Silicon is widely used

as a semiconductor because it remains more of a semiconductor at higher

temperatures than another material of the same family, germanium, and

also because its native oxide is easily grown in a furnace and forms a good

semiconductor/dielectric interface. In the form of silica and silicates, sili-

con comprises useful glasses, cements, and ceramics. It is also a constituent

of silicones, a class-name for various synthetic plastic substances made of

silicon, oxygen, carbon and hydrogen. In naturally found Si and also for

the industrial uses one often comes across defects in crystalline structure of

the material and hence the study to know more about the pros and cons of

those defects came into practice. There are several phenomena such as the

electrical transport and phototransport, the light absorption and emission

etc., for which the investigation of defects is essential. Sometimes the de-

fects are useful and sometimes they have to be avoided because they make

troubles. In bulk semiconductors, there are various specific applications like

the radiation detectors where the defects are very useful. Since a long time

6

Silicon is used in radiation detectors and electronic devices. Nowadays, these

devices work on submicron technology and they are parts of integrated cir-

cuits with large to very large scale integration. Silicon and silicon based

devices are heavily worked upon in great theoretical and experimental de-

tail, in many fields of physics including particle physics experiments, nuclear

medicine, reactors and space. Defects in the material represent a limiting

factor in the operation of devices. In spite of the huge research being put

into by the scientific community till date, there are a lot of aspects not clari-

fied, related to the behavior of impurities, defects, vacancies in Silicon. And

hence a global understanding of their local structure and properties became

increasingly important due to the reduction in chip sizes and to the increase

of the operation speed. The study of effects of point defects on electronic,

structural and vibration properties of bulk semiconductors, and also on low

size semiconductor structures is a thematic of high interest. This study will

bring contributions to the fabrication of Si based smart materials for photon-

ics, MOS and nanowires based devices etc. The fabrication of new materials

will open up new horizons and enable path-breaking advances in science and

technology.

In bulk semiconductors various defects (sometimes called trap defects), like

the point defects, impurities or local stresses, are located in the volume of

the material. In nanocrystalline semiconductors the trapping phenomena are

dominated by the traps located at the surface or interface, due to the very big

area/volume ratio (of the order of 108m–1 for nanocrystals). These traps are

produced by the adsorption, dangling bonds, and the internal stresses (in-

duced by misfit)[1]. Experimental methods like the photo-induced current

transient spectroscopy (PICTS), the thermally stimulated currents (TSC),

the thermally stimulated depolarization currents (TSDC), and the optical

charging spectroscopy (OCS) are in practice to study those defects. In the-

ory as well scientists are making headway through the calculations of the

stability, energetics of formation, migration, electrostatics etc. of all such

defects. Before going into the further discussion about the defects and their

respective analyses, one may feel the need to classify broadly the types of

defects that are common in Silicon.

7

1.2 Types of defects

In reality there is no existence of a perfect crystal. All crystals have some

defects. These defects do in general contribute to various mechanical and

electronic properties of the material. In fact a part of modern electronic

engineering is based on the manipulation of these defects for efficient use of

the material. Grossly these defect types in silicon crystals can be grouped as

following :

• Point defects. (vacancies, interstitials, impurities etc.)

• Linear defects (dislocations etc.).

• Planar defects (grain boundaries, stacking faults etc.)

1.2.1 Point defects

When in the lattice structure of a material, an atom is missing or is in an

irregular place, then that material is said to have point defects. This class

of defects includes self interstitial atoms, interstitial impurity atoms, substi-

tutional impurities and vacancies. A self interstitial atom is an extra atom

that has crowded its way into an interstitial void in the crystal structure.

A substitutional impurity atom is an atom of a different type than the bulk

atoms, which has replaced one of the bulk atoms in the lattice. Interstitial

impurity atoms are much smaller than the atoms in the bulk matrix and

they are found to fit in the inter-atomic space between the bulk atoms of the

lattice structure. Vacancies are basically the empty spaces where an atom

should be, but is missing. They are common in metals and semiconductors,

especially at high temperatures when atoms are found to change their posi-

tions pretty often leaving behind empty lattice sites. In most cases diffusion

(mass transport by atomic motion) can only occur because of vacancies. The

point defects are also responsible for the lattice strain in the crystal because

of the deformation in the geometry around the defect itself. In the adjoining

Figure 1.1, an example of the occurence of all the point defects discussed

here, is schematically described. The vacancies can be charged giving rise

8

Figure 1.1: Various point defects in the bulk material.

to a special sub-branch of defects known as the charged defects. Even doping

the host material with foreign atoms (technically impurities) produce charged

defects. The cases of doped Silicon crystals can be considered as an example,

where donor atoms provide excess electrons to form n-type silicon and accep-

tor atoms provide a deficiency of electrons to form p-type silicon. It is also

worth mentioning here that combination of the point defects, eg. a vacany

+ an interstitial can give rise to another category of charged defects. Those

defects can influence the formation of cluster defects in the bulk material.

In this regard it is worth mentioning a special type known as the Frenkel

defects. A Frenkel defect, Frenkel pair, or Frenkel disorder is a type of point

defect in a crystal lattice where the defect forms when an atom or cation

leaves its place in the lattice, creating a vacancy, and becomes an interstitial

by lodging in a nearby location not usually occupied by an atom. Frenkel

defects occur due to thermal vibrations, in principle there will be no such

defects in a crystal at 0 K.

9

1.2.2 Linear defects

These defects are found in the material when an array of atoms is displaced

in the crystal structure. The presence of such dislocations affect the physcial

properties of the material to a pretty large extent, in fact the motion of the

dislocations are the cause for the plastic deformations to occur in the ma-

terial. The dislocations can be broadly devided into Edge dislocations and

Screw dislocations. An edge dislocation is said to have occured where an

extra half-plane of atoms is introduced half way in the middle through the

crystal, disrupting the symmetry of nearby planes of atoms. The disloca-

tion is called a line defect because the locus of defective points produced

in the lattice by the dislocation lie along a line. The inter-atomic bonds

are significantly distorted only in the immediate vicinity of the dislocation

line. The screw dislocation is not very straightforward to understand. The

motion of a screw dislocation is also a result of shear stress, but the de-

fect line movement is perpendicular to direction of the stress and the atom

displacement, rather than parallel. Figure 1.2 describes the two different

dislocations schematically. The edge dislocations allow deformation to occur

Figure 1.2: Edge and Screw dislocations.

at much lower stress than in a perfect crystal. One can understand that

10

fact by studying the movement of the edge dislocation. Dislocation motion

is analogous to movement of a caterpillar. A schematic diagram in Figure

1.3 shows the movement of the dislocation. As for the screw dislocations the

Figure 1.3: The movement of the edge dislocation.

dislocations move along the densest planes of atoms in a material, because

the stress needed to move the dislocation increases with the spacing between

the planes. FCC and BCC metals have many dense planes, so dislocations

move relatively easy and these materials have high ductility.

1.2.3 Planar defects

Defects like the Stacking faults, Twin boundaries, Grain boundaries are ex-

amples of planar defects. A stacking fault is a one or two layer interruption

in the stacking sequence of atom planes. Stacking faults occur in a number

of crystal structures. For example we can consider the faults of hcp and fcc

structures. Here the first two layers arrange themselves identically, and are

said to have an AB arrangement. If the third layer is placed so that its atoms

are directly above those of the first (A) layer, the stacking will be ABA. This

indeed is the hcp structure, and it continues ABABABAB. However it is

possible for the third layer atoms to arrange themselves so that they are in

line with the first layer to produce an ABC arrangement which is that of

the fcc structure. So, if the hcp structure is going along as ABABAB and

suddenly switches to ABABABCABAB, there is a stacking fault present.

Another type of planar defect is the grain boundary. The interface between

11

Figure 1.4: (a) Photograph showing a model of the ideal packing in the ABFCC structure, (b) Photograph showing a model of the AB FCC structurewith a stacking fault (c) Differently oriented crystallites in a polycrystallinematerial forming boundaries.

two grains, or crystallites, in a polycrystalline material is known as the grain

boundary. Grain boundaries are such defects in the crystal structure which

are found to decrease the electrical and thermal conductivity of the material.

Grain boundaries limit the lengths and motions of dislocations. It is known

now that having smaller grains (more grain boundary surface area) strength-

ens a material. Different experimental procedures can be used to control the

size of the grains.

1.3 A brief history of defects in bulk Si

Let us now focus on the study of defects in crystalline Silicon. Various exper-

iments and theoretical calculations have revealed some interesting properties

of the defects. In both forms of study there are frontiers which are difficult to

conquer, mostly because of the complexity of the systems. In some cases the

experiments are very difficult to carry out, whereas in some cases to construct

the real theoretical picture of the system is extremely challenging. Some facts

which will be mentioned in the following section are well researched and es-

tablished. As for example we have known that the stability of crystalline

silicon comes from the fact that each silicon atom can accommodate its four

valence electrons in four covalent bonds with its four neighbors. The pro-

12

duction of primary defects or the existence of impurities or defects destroys

this fourfold coordination. It has been established[3][4] that the structural

characteristics of the classical vacancy in bulk Si are: the bond length in

the bulk is 2.35 A and the bond angle –109 degrees . The formation energy

is 3.01 eV (p-type silicon), 3.17 eV (intrinsic), 3.14 eV (n-type). For inter-

stitials, different theoretical works have reported the possibility of various

structural configurations. They are a) the hexagonal configuration, a sixfold

coordinated defect with bonds of length 2.36 A, joining it to six neighbors

which are fivefold coordinated; b) the tetrahedral interstitial is fourfold co-

ordinated; has bonds of length 2.44 A, joining it to its four neighbors, which

are therefore fivefold coordinated; c) the split − < 110 > configuration: two

atoms forming the defect are fourfold coordinated, and two of the surround-

ing atoms are fivefold coordinated; d) the ‘caged’ interstitial contains two

normal bonds, of length of 2.32 A, five longer bonds in the range 2.55 2.82

A, and three unbounded neighbors at 3.10 3.35 A. The calculations [6]-[8]

found that the tetrahedral interstitial and caged interstitial are metastable.

For interstitials, the lowest formation energies (calculated theoretically) in

eV are 2.80 (for p-type material), 2.98 (for n-type) and 3.31 in the intrinsic

case respectively. Authors have reported that in silicon the vacancy takes on

five different charge states in the band gap, viz. V 2+, V +, V 0, V − and V 2–[9].

But about the charge states of the self-interstitial there are differences of

opinions in reports by Lopez[10] et. al. The experimental examination of

primary point defects buried in the bulk is difficult and for various defects

this is usually indirect. In a series of theoretical studies[2] and correlated

EPR and DLTS experiments of Watkins and co-workers[4], it became possi-

ble to solve some problems associated with the electrical level structure of the

vacancy. In crystalline silicon bombarded with energetic projectiles, the di-

vacancy center is being studied for quite some time now by numerous authors

applying various experimental techniques, e.g. EPR [3], photoconductivity

[11], infrared absorption [12], electron-nuclear double resonance (ENDOR)

or deep level transient spectroscopy (DLTS) and at the room temperature it

is considered as a stable defect. The undisturbed configuration of divacancy

could be viewed as two vacant nearest neighbor lattice sites. Experiments by

13

Watkins et. al. with isochronal and isothermal annealing studies reveals that

the divacancy anneals out at 570 K[3]. Since then many other experiments

have tried to comment on the formation mechanism and charge states of the

divacancy.

In 2002, in an important report, S. Goedecker, T. Deutsch and L. Billard

[13] predicted the existence of a new type of primary defect in silicon and

thus a new type of symmetry in the crystallography of the material. Their

work proved the existence of a fourfold coordinated defect which is found to

be more stable. Experiments by Lazanu and Lazanu [14] proved the exis-

tence of such defects. Lattice Monte Carlo calculations are done by Damien

Caliste and Pascal Pochet[15] to study the diffusion of vacancies in bulk Si.

They have reported the results of simulations of vacancy-assisted diffusion

in silicon to show that the observed temperature dependence for vacancy

migration energy is explained by the existence of three diffusion regimes for

divacancies. The geometry and energetics of the normal divacancies and split

divacancies are discussed as well. In a follow up work Damien et. al. have

presented an analysis of stress-enhanced vacancy-mediated diffusion in biax-

ially deformed Si (100) films as measured by the strain derivative (Q′) of the

activation energy[16].

1.4 The Motivation

There are in fact two parts of this thesis. The first part deals with the

accuracy of Hartwigsen-Goedeker-Hutter (HGH) pseudopotentials with Non

Linear Core Correction (NLCC). The main idea is to use the pseudopoten-

tials for further accurate calculations. One can argue of course the validity

of such a job. We have tried to find out a less complex approach with

the norm-conserving pseudopotentials to produce as accurate results as that

of the Projector Augmented Wave[80] (PAW) approach or that of the all-

electron calculations. The success of our approach would ensure a method

other than the PAW method for accurate calculations of various physical

quantities. This is an attempt to correct the otherwise implied linearization

of the pseudopotential. Chapter 4 would deal in detail with this part of the

14

research. For the next and more important part of the research we wish

to concentrate ourselves in the charged point defects in silicon nanoclusters.

Up to now, as will be explained in Chapter 3, the charged defects have been

simulated within the framework of Periodic Boundary Conditions (PBC).

Indeed this PBC approach is good to simulate an infinite bulk. But, as will

be shown in detail in the following chapters, this PBC approach comes with

its own artifacts which can be difficult to deal with. The electrostatics, in

particular for those systems is not simple to represent and a lot of mathe-

matical jargon is needed to account for some other unwanted interactions.

In our approach we have tried to find a method with which we can deal with

isolated charged systems and also can comment on bulk properties even when

we are working with Free Boundary Conditions (FBC). Our motivation is to

find the correct electrostatics of the charged defects in the nanoclusters and

its ability to extrapolate those results for the infinite bulk cases. Chapter 5

deals with this part of the research in detail. Before going into the details

of the current research work it is handy to have an overview of some basic

theoretical concepts.

15

Chapitre 2

Ce chapitre traite des differents concepts theoriques a la base de cette

these. Une vue generale de la theorie de la fonctionnelle de la densite (DFT)

dans ces differents aspects est decrite. J’aborde aussi les differentes fonction-

nelles d’echange-correlation (ECF) d’interet pour la these, ainsi que la theorie

sous-jacente. La these utilisera par la suite une fonctionnelle GGA. Enfin,

je traite aussi les differentes bases existantes pour resoudre les equations

de Kohn-Sham (KS), montrant ainsi l’interet des ondelettes pour le traite-

ment des defauts. Tous les calculs ayant ete fait avec l’approximation des

pseudo-potentiels, la theorie est abordee ici. La these est basee sur la forme

des pseudo-potentiels de Hartwigsen-Goedecker-Hutter (HGH) pour le calcul

des defauts aborde au chapitre 5, et les HGH avec correction non-lineaire de

cœur pour le calcul des energies d’atomisation decrites dans le chapitre 4.

16

Chapter 2

Some Theoretical Concepts :

DFT and its components

2.1 Introduction

In this chapter we discuss in order of appearance the basic theoretical con-

cepts of Density Functional Theory, along with its major components - the

Pseudopotentials and the Exchange Correlation Functional. The different

basis sets which can be used to solve the Kohn-Sham equations are also

described with their respective pros and cons.

2.2 Density functional theory (DFT) (a brief

overview)

Since its discovery almost four decades back, DFT, a theory of electronic

ground state structure, has been one among the most important tools to

understand the ground state properties of molecules, solids and clusters.

Theoretical physicists and chemists find in it an altrenative to the traditional

methods of quantum chemistry dealing with many-electron wave-function.

DFT in its full glory is a thematic continuation of the previous approximate

methods although itself being exact in principle. With a motive to reduce the

number of parameters needed to describe the many body system, DFT works

17

centering around the electron density ρ(r) of the electronic system. The main

goal of almost all physical systems is to know its various physical properties

and structure, and hence calculation of the total energy of the system is most

important. Keeping in accordance with that goal, all the contributions to the

total energy are expressed in terms of the electron density in the framework

of DFT. In the following subsections various important structural parts of

the DFT formalism are explained.

2.2.1 Hohenberg-Kohn theorems

Among the structural pillars of DFT Hohenberg-Kohn (HK) theorems [17]

are of foremost importance. The first HK theorem proves that the ground

state electron density ρ is sufficient to determine , in principle, the total

energy of the system and hence any ground state property of the system

without the knowledge of the many electron wave function can thus be ob-

tained. The second theorem states that the ground state energy of the system

is the minimal value of the energy functional of the electronic system. Both

the two theorems are proven here in the following subsections.

First Hohenberg-Kohn theorem. The ground state density ρ (r) of a

system of many interacting electrons moving in some external potential v(r)

determines this potential uniquely.

Proof. [18] By reductivo ad absurdum. Let us consider two external potentials

v1(r) and v2(r) having the same ground state electron density ρ(r), with

Hamiltonians H1 and H2 respectively, and non-degenerate ground states |Ψ1〉and |Ψ2〉, therefore

H1 |Ψ1〉 = E01 |Ψ1〉 (2.1)

H2 |Ψ2〉 = E02 |Ψ2〉 . (2.2)

Let us apply the variational principle at this point

E01 < 〈Ψ2|H1 |Ψ2〉 = 〈Ψ2|H2 |Ψ2〉+ 〈Ψ2|H1 −H2 |Ψ2〉 (2.3)

< E02 +

∫ρ(r) [v1(r)− v2(r)] dr . (2.4)

18

Evidently by interchanging subscripts 1 and 2 in the above inequality another

inequality can be formed of similar structure. By adding both inequalities

we can get

E01 + E0

2 < E02 + E0

1 (2.5)

which is obviously not true. Hence by contradiction to our initial assumption

we can conclude that there is one and only one external potential

associated with the ground state electronic density.

Since ρ(r) is known for the system, all other ground state properties like

the particle number, and the external potential, the total energy E [ρ], the

kinetic energy T [ρ], the potential energy V [ρ], etc. can be calculated directly.

Second Hohenberg-Kohn theorem. If any electron density, given by,

ρ(r) is such that∫ρ(r)dr = N and ρ(r) > 0 then

E0 6 Ev [ρ] = T [ρ] + Vne [ρ] + Vee [ρ] ≡∫

v(r)ρ(r) dr+ FHK [ρ] . (2.6)

where E0 is the ground state energy, Vne and Vee are potentials due

to nucleus-electron and electron-electron interactions. FHK is the univer-

sal functional as it describes a treatment of the kinetic and internal potential

energies which is same for all systems.

Proof. According to the first HK theorem any electron density ρ(r) uniquely

determines the external potential v(r) consequently with its own Hamiltonian

and the ground state wave function 〈Ψ| associated with this Hamiltonian. Let

〈Ψ| describe the ground state of a system with density ρ and Hamiltonian

H. According to the Rayleigh-Ritz variational principle, for any electronic

state 〈Ψ| ⟨Ψ∣∣∣H

∣∣∣Ψ⟩≥ 〈Ψ|H |Ψ〉 = E [ρ] (2.7)

and ⟨Ψ∣∣∣H

∣∣∣Ψ⟩=

∫v(r)ρ(r) d3r+ T [ρ] + Vee [ρ] = Ev [ρ] . (2.8)

Therefore

Ev [ρ] ≥ Ev [ρ] (2.9)

19

2.2.2 The Kohn-Sham (KS) equations

In the framework of DFT KS equations [18] are essentially Schroedinger

equations of a system of noninteracting particles that would generate the

same density as one given system of interacting particles. To that end we

need to derive the equations for the single particle system that yields the

density ρ(r). To start with let us focus in deriving the equations for a system

of noninteracting particles. For a noninteracting system, the total energy of

the system can be written as

E[ρ] = Ts [ρ] +

∫v(r)ρ(r) , (2.10)

where v is the net potential and Ts [ρ] is the kinetic energy of the ground

state of the system of noninteracting electrons having a density ρ(r). The

density of the system is given by

ρ(r) =N∑

i=1

|ϕi(r)|2 . (2.11)

The number of occupied states N being constant, let us consider the variation

in density as

δρ(r) =N∑

i=1

δϕi∗(r)ϕi(r) (2.12)

Since the density in equation 2.11 is stationary with respect to the given

variation (the total number of particles being constant), the integral of the

variation of density is evidently zero.

∫δρ(r)dr =

∫ N∑

i=1

δϕi∗(r)ϕi(r)dr = 0 . (2.13)

Let us now apply the trick of Lagrange multiplier. For this case with the

lagrangian multiplier ǫ, the variation in the energy functional takes the form

20

as

δE[ρ(r)] =

∫δρ(r)

v(r) +

δ

δρ(r)Ts[ρ(r)]− ǫ

dr = 0 . (2.14)

The system we have at hand is of noninteracting particles. Hence we can

write the ground state energy eigenvalue of the system as

E =N∑

i=1

ǫi (2.15)

and obtain as single-particle equations

[−1

2∇2 + v(r)− ǫi

]ϕi(r) = 0 (2.16)

The objective is to find equations for the states ϕi, but it must be kept in

mind that these equations (2.16) are valid only for noninteracting particles.

Obviously they yield the density ˜ρ(r) of a system of noninteracting particles.

Whereas in reality the quantity we are interested in is the density ρ(r) of

a system of interacting particles. In the framework of DFT equation (2.10)

needs to be rewritten as

E[ρ] = FHK [ρ] +

∫vext(r)ρ(r) , (2.17)

where vext is the external potential and FHK [ρ] is the density functional,

which we can split into expressions like

FHK [ρ] = Ts [ρ] +1

2

∫ ∫ρ(r)ρ(r′)

|r− r′| drdr′ + EXC [ρ] , (2.18)

In 2.18, T [ρ] is the noninteracting part of the kinetic energy of this many

electron system we are dealing with

Ts [ρ] =∑

i

〈ϕi| −1

2∇2

r|ϕi〉 . (2.19)

The second term in equation 2.18 describes the classical electrostatic inter-

action between the noninteracting particles. This is the Hartree energy of

21

the system. EXC [ρ] is the exchange-correlation energy, which takes in ac-

count the entire electron-electron interaction effect beyond the Hartree term

viz. the exchange energy due to the Pauli exclusion principle, the correla-

tion energy and the difference in kinetic energy between an interacting and a

noninteracting system. In practice the second and third term are collectively

considered with the external potential as the effective Kohn-Sham potential

veff (r, ρ(r)) = vext(r) +

∫ρ(r′)

|r− r′|dr′ +

δEXC [ρ(r)]

δρ(r). (2.20)

The exchange-correlation potential is described by the third term as

vxc(r) =δExc [ρ]

δρ. (2.21)

Hence equation 2.17 can alternatively be written as

E[ρ] = Ts [ρ] +

∫veff (r)ρ(r) , (2.22)

The particles described by equation 2.22 and 2.10 are of course different yet

since those two equations are essentially similar they will yield similar single

particle equations. Finally we can write the most appropriate form of the

Kohn-Sham equations as

[−1

2∇2 + veff (r)

]ϕi = ǫiϕi , (2.23)

ϕi and ǫi being the single particle orbitals and energies. At this point

it is pretty clear that the ground state electron density can be computed

from the self-consistent solution of the KS equations as in 2.23 only if the

expression for the exchange-correlation potential is known. In practice, the

density is calculated with the formula in equation 2.24 and consequently

the obtained density is inserted in the exchange-correlation potential as in

equation 2.21. Then the new eigenstates and new electron density of the

22

system are calculated, until the required convergence is reached.

ρ(r) =N∑

i=1

|ϕi(r)|2 (2.24)

It is noteworthy here that the KS equations as in equation 2.23 yield eigen-

states ϕi of fictitious particles which only have the same density as the real

particles. Hence it must be kept in mind that the ground state energy of the

real system is not the simple sum of the Kohn-Sham single-particle eigen-

values as for noninteracting particles 2.15. Let us now have a look at the

ground state energy (of the real system). It would look like

E =N∑

i=1

ǫi + Exc[ρ(r)]−∫

vxc(r)ρ(r)dr−1

2

∫ρ(r)ρ(r′)

|r− r′| drdr′. (2.25)

KS DFT is a technique to describe the ground state energy and density, more

to say almost all ground state properties of a many body system in terms

of single particle equations and states. It is trap that one may find himself

in if one believes, that in this formalism, real electrons are being described

as independent particles experiencing an external field of the ions and of

all other electrons. In reality the KS equations are merely the equations of

noninteracting fictitious particles, whose density is the same as the density of

real electrons in the ground state. Hence it can be said that the eigenstates

of the KS equations do not have a direct physical meaning. However, in a

system of N occupied states the N-th KS eigenvalue, ǫN , in equation (2.23)

finds one a physical interpretation [20]

• The ionization potential (IP) of a finite system is given by

ǫN = −IP (2.26)

• And the same is the chemical potential for extended systems µ

ǫN = µ (2.27)

23

It depends on the approximation used for the exchange-correlation func-

tional/potential how close ǫN comes to the exact ionization potential or

chemical potential [21].

2.3 Different Basis Set Approaches

As one can expect there are different approaches to solve the KS equations.

The approaches are based on the basis sets on which this KS equations are

expanded and then solved accordingly. Within this chapter I would discuss

the three main approaches in terms of basis sets viz. the localized basis set

approach, the plane wave approach and the wavelet basis set approach.

2.3.1 Atomic Basis Sets

Since the first efforts to solve the KS equations the use of atomic basis sets

has been serving the cause to a very good effect. In this approach the molec-

ular orbitals (MO) are expressed as a linear combination of atomic orbitals

(LCAO). The basic strength of the LCAO approach is its general applicabil-

ity as in reality it can work on any molecule with any number of atoms. To

take the LCAO concept another step ahead to facilitate the calculations we

can use a larger number of atomic orbitals (AO)(e.g. a hydrogen atom can

have more than one s AO, and some p and d AOs, etc.). This helps us to

achieve a more flexible representation of the MOs and therefore more accu-

rate calculated properties according to the variation principle. One can also

use AOs of a particular mathematical form that simplifies the computations

(but are not necessarily equal to the exact AOs of the isolated atoms). These

AOs are called the basis functions, or more precisely the localized atomic ba-

sis functions. Instead of having to calculate the mathematical form of the

MOs (impossible on a computer) the problem is reduced to determining the

MO expansion coefficients in terms of the basis functions. Functions that

resemble hydrogen AOs (Slater functions in other words) are very suitable

for expanding MOs because they have the correct shape (a) near the nucleus

(shape of a cusp) and also (b)far from the nucleus (decay like exp−ar). The

24

main reason for the preference of the Gaussian functions is the fact that they

allow for efficient computation of molecular integrals. In quantum chem-

Figure 2.1: Illustration of the simplicity of the Gaussian functions over theslater functions

istry terminology a single Gaussian function is called a primitive Gaussian

function, or primitive GTO (Gaussian Type Orbital). Some programs use

cartesian primitive GTOs while the others use spherical primitive GTOs. In

mathematical terms spherical and cartesian functions are the same for up

to l = 1 (p functions) but differ slightly for l = 2 or higher. In practice,

fixed linear combinations of primitive Gaussian functions are used which are

called Contracted Gaussians (CGs). The simplest kind of CGs are the STO-

nG basis sets . These basis sets attempt to approximate Slater-type orbitals

(STOs) by n primitive Gaussians. The STO-nG basis sets are not that sat-

isfactory as they include only one CG per atomic orbital. Improved basis

sets are obtained by including more than one CG per atomic orbital, e.g.:

DZ (“double zeta”), TZ (“triple zeta”), QZ (“quadruple zeta”). Or the im-

provement can also be achieved by the use of one CG per core atomic orbital

and more than one for the valence atomic orbitals, e.g. SV, 3-21G, 4-31G,

6-31G, 6-311G. Increasing the number of CGs per atomic orbital will not

25

actually give a good quality basis set because in reality there are other types

of CGs which are to included. For example one can include CGs of angular

momentum higher than in the valence orbitals of each atom. These polar-

ization functions enhance the flexibility of atoms to form chemical bonds in

any direction and hence improve the calculated molecular structures. Few

examples of such type are DVP, TZP, cc-pVDZ, cc-pVTZ etc. One can also

take in account the CGs which extend further from the nucleus than the

atomic orbitals. Such diffuse functions improve the predicted properties of

species with extended electronic densities such as anions or molecules form-

ing hydrogen bonds. Basis sets are considered balanced when they include

both polarization and diffuse functions. Examples of these types comprises

6-31+G*, 6-311++G**, aug-cc-pVDZ etc. But completeness for these types

of basis sets also calls for it to be infinite which is computationally unachiev-

able. Indeed the atomic basis sets give out the real physics of the molecules

but the two point integrations are exceedingly difficult to calculate with such

basis sets. Again there is a problem of physical convergence of the computed

results with these basis sets. In principle more informations can be put in to

the computational structure but to need the complete picture what we need

is an infinite basis set of this kind. Which of course is not possible to realize

in reality.

2.3.2 Plane Wave Basis Sets

This approach is one where the Fourier representation of the equations bear a

heavy significance. In reality most computational operations are easy to work

with in the Fourier space which hands over an advantage to the plane wave

basis sets in DFT calculations. According to the Bloch’s theorem of solid

state physics, the electronic wave functions at each k point can be expanded

in terms of a discrete plane wave basis set. In principle, an infinite plane-wave

basis set is required to expand the electronic wave functions. However, the

coefficients for the plane waves with small kinetic energy are more important

than those with large kinetic energy. Thus one can understand that there is a

possibility for the plane-wave basis set to be truncated to include only plane

26

waves that have kinetic energies less than some particular cutoff energy. The

basis set would be infinite if a continuum of plane-wave basis states were

required to expand each electronic wave function, irrespective of the small

magnitude of the cutoff energy. Thus one feels a need to make this apparently

infinite basis set to a finite one. Application of the Bloch theorem allows

the electronic wave functions to be expanded in terms of a discrete set of

plane waves. To that end if one introduces a cutoff energy to the discrete

plane-wave basis set we can have a finite basis set. Hence one can say that to

achieve completeness plane wave basis sets should be infinite as well but with

the cutoff approximation appropriately accurate results can be obtained. An

error will be there for sure in the computed total energy due to the truncation

of the plane-wave basis set at a finite cutoff energy. However, it is possible

to reduce the magnitude of the error by increasing the value of the cutoff

energy. In principle, the cutoff energy should be increased until the calculated

total energy has reached convergence. One of the difficulties associated with

the use of plane-wave basis sets is that the number of basis states changes

discontinuously with cutoff energy. Generally these discontinuities are found

to occur at different cutoffs for different k points in the k-point set. Another

important aspect is that at a fixed-energy cutoff, a change in the size or shape

of the unit cell will cause discontinuation in the plane-wave basis set. This

problem is generally taken care of by using denser k-point sets, so that the

weight attached to any particular plane-wave basis state is reduced. However,

the problem is still present even with quite dense k-point samplings. It can

be handled by applying a correction factor which manages to account for

the difference between the counts approximately number of states in a basis

set with infinitely large number of k points and the number of basis states

actually used in the calculation [56]. In thsi approach though one needs to

compute the coefficients which are in general large in number. This aspect

raises a question about the speed of the process for calculations of very large

systems. In reality this is an order N method. Very large systems can be

tackled if we can use the plane wave DFT codes with parallel computation.

But it is very difficult to parallelize the plane wave computations mainly

because of the complexities raised by Fast Fourier Transformation (FFT).

27

2.3.3 Wavelet Basis Sets

A novel approach to solve the KS equations were shown to be very suc-

cessful by Genovese and his colleagues[43]. They have used the Daubechies

wavelets as a powerful systematic basis set for electronic structure calcula-

tions and have shown that they are good because they are orthogonal and

localized both in real and Fourier space. Daubechies wavelets have been

found to have all the properties that a DFT program would like to have of a

basis set used for the simulation of isolated or inhomogeneous systems. They

form a systematic orthogonal and smooth basis, localized both in real and

Fourier spaces and that allows for adaptivity. Hence a DFT approach based

on such functions will be very good to satisfy the requirements of precision

and localization found in many applications. A wavelet basis comprises a

family of functions generated from a mother function and its translations

on the points of a uniform grid of spacing h. Here the number of basis

functions is increased by decreasing the value of h (which is an analog to

the cutoff energy for the case of plane wave basis sets). Because the basis

set in systematic , the numerical description is reported to be more precise.

The degree of smoothness determines the speed with which one converges

to the exact result as h is decreased. The degree of smoothness increases as

one goes to higher order Daubechies wavelets. In the method used for the

DFT code BigDFT (the one which is used for the calculations in this thesis)

Daubechies wavelets of order 16 are used. A very high rate of convergence

is achieved in comparison to other finite difference, finite element, or real

space methods[57][58][59]. Luigi et. al. have discussed the need of the lo-

calization of the basis sets in real space as essential for molecular systems.

This is becaude for basis sets that are not localized in real space are waste-

ful in the context of molecular systems. For example, with plane waves one

has to fill an orthorhombic cell into which the molecule fits. There can be

regions of the cell that may contain no atoms and hence no charge density.

But the plane wave approach cannot be used to utilize this scenario. Since

Daubechies wavelets have a compact support, one can consistently define a

set of localization parameters, which allows the researchers to put the basis

28

functions only on the points that are very close to the atoms. The compu-

tational volume in this method is thus given only by the union of spheres

centered on all the atoms in the system. Real space localization is also nec-

essary for the implementation of linear scaling algorithms. Hence this family

of basis set is a worthy contender for developing such algorithms. A technical

fact can be looked into at this point. For a given system, the convergence

rate of the minimization process depends on the highest eigenvalue of the

Hamiltonian operator. Since the high frequency spectrum of the Hamilto-

nian is dominated by the kinetic energy operator, high kinetic energy basis

functions are therefore also approximate eigenfunctions of the Hamiltonian.

A function localized in Fourier space is an approximate eigenfunction of the

kinetic energy operator. By using such functions as basis functions for the

KS orbitals, the high energy spectrum of the Hamiltonian can thus easily

be preconditioned. It is of course known that a high degree of adaptivity

is necessary for all-electron calculations since highly localized core electrons

require a much higher spatial resolution than the valence wave function away

from the atomic core. This fact is also discussed in the previous paragraphs.

The important fact is that pretty high adaptivity can be obtained with a

wavelet basis. In BigDFT use of pseudopotentials is undertaken to a very

good effect. This is because such pseudopotentials are the easiest way to in-

corporate the relativistic effects that are important for heavy elements. The

use of pseudopotentials readily reduces the need for adaptivity and one has

therefore only two levels of adaptivity. Now one will have a high resolution re-

gion that contains all the chemical bonds and a low resolution region further

away from the atoms where the wave functions would decay exponentially

to zero. In the low resolution region each grid point carries a single basis

function. In the high resolution region it carries in addition seven wavelets.

In terms of degrees of freedom, the high resolution region is therefore eight

times denser than the low resolution region. In comparison with a plane

wave method this wavelet method is therefore particularly efficient for open

structures with large empty spaces and a relatively small bonding region.

29

2.4 Approximations to the Exchange-Correlation

Potential

In this section we talk about the approximation which is inherent to the DFT

methods. It is the approximation to the exchange-correlation potential.

In equation (2.21)the exchange correlation potential is defined as

vxc(r) =δExc [ρ]

δρ. (2.28)

Although the KS equations appear very promising yet without the explicit

knowledge of the exchange correlation functional the KS equations cannot

be solved exactly. Among the cons of DFT finding a good approximation

to the exchange correlation potential is of prime importance. In this section

two very important approaches for the approximation to the potential are

discussed.

2.4.1 The Local Density Approximation (LDA)

In the framework of LDA [19] the electron density of local area of the in-

homogeneous system is approximated with the same density as that of a

homogeneous electron gas in the same extremity. If we take ǫxc to be the

exchange correlation energy per electron of a homogeneous electron gas of

density ρ, then the real exchange correlation energy functional can be ap-

proximated by

Exc[ρ] ≈ ELDAxc [ρ] ≡

∫ρ(r)ǫxc(ρ(r)) dr . (2.29)

the expression for vxc can be obtained by taking the functional derivative of

Exc with respect to ρ

vLDAxc =

δExc

δρ= ǫxc(ρ) + ρ

∂ǫxc∂ρ

(2.30)

30

and, since the system is not homogeneous, ρ = ρ(r). It is assumed hereby

that ǫxc can be calculated accurately for the homogeneous electron gas. In

fact, ǫxc can be divided into exchange and correlation parts ǫxc = ǫx + ǫc.

The exchange part is known analytically and given by

ǫx =3

4

(3ρ(r)

π

)1/3

(2.31)

and the LDA exchange potential is given as

vx(r) = −(3

πρ(r))1/3 , (2.32)

while for the correlation part one has to fall back on the Perdew-Zunger

parametrization of quantum Monte-Carlo data for the electron gas [23]. It is

obvious from the discussion till here that for a homogeneous electron gas, the

LDA exchange-correlation functional is exact. Despite the fact that for most

applications, especially for isolated systems, the electron density differs by a

fair margin from that of a homogeneous electron gas, the approximation still

produces acceptable and good results. This can be understood from the ex-

planation that the LDA satisfies the sum rule which expresses normalization

of the so-called exchange correlation hole [18]. In other words, given that

an electron is at the position r, the electron density for the other electrons

is depleted near r. A ’hole’ in the density distribution ρ(r′). is dug by the

electron at position r due to Pauli principle and electron-electron interaction.

It is thus normalized as

∫ρxc(r, r

′)dr′ = −1 (2.33)

2.4.2 Overview of the performance of LDA

In a report[29] results showing the success of LDA formalism have listed.

In a thumbnail representation, the success story of LDA can be tabulated as

the following,

• Binding energies are often better than 1 eV but in some s–d bonded

31

systems the error can be twice or even three times as large. There is a

systematic over binding.

• Equilibrium distances are generally accurate to within 0.1 A, They are

systematically too short. Hence in general it can be said that geometries

are accurately computed within this framework.[22]

• Vibrational frequencies are accurate to within 10–20 %. There exist

occasional cases with larger errors.

• Charge densities are better than 2 %.

• LDA results are nearly always much better than those of the Hartree–Fock

(HF) approximation.

As for the validity of the above observations we can have a look at the follow-

ing tables in figure 2.2 taken from the works of Muller, Jones and Harris[30].

The results shown here, however, also indicate some problematic deficiencies.

Most notable is the systematic overbinding predicted by the LDA, particu-

larly for the s-d bonded systems. Although the overbinding is to a much

lesser extent yet is reflected in a small but relatively systematic underesti-

mate of the bonding distances. The following small list gives instances of

systems for which the LDA poses some serious problems.

• The transition-metal oxides FeO and CoO are erroneously predicted

to be metallic. But, MnO and NiO come out as anti-ferromagnetic

insulators in accordance with experiment[31].

• Solid Fe is predicted to be an fcc paramagnet [32] but is a bcc ferro-

magnet at low temperatures.

• In many semiconductors, the LDA gives the metal-insulator transition

at much too large volumes[33].

• The LDA predicts the wrong dissociation limits for a large number of

molecules[34].

• The LDA predicts incorrect ground states for many atoms[35].

32

Figure 2.2: Ground-state properties of the molecules H2O, NH3, and CO2,as obtained from the LDA and from experiment. It is assumed that thenumerical errors involved in obtaining the LDA results are negligible in com-parison to the deviation between theory and experiment. The equilibriumdistances in Table II are probably exceptions to the assumptions becausethey do not conform to the general expectations of bond distances being tooshort within the LDA.

• The LDA gives unstable negative atomic ions in many cases when these

are stable[35].

2.4.3 Generalized Gradient Approximations (GGA)

As stated in the previous section, we can understand that although LDA

is quite adequate for some cases, yet for most systems a higher accuracy is

desired. An error in a binding energy of the order of 20 % or 1 eV is, e.g.,

not acceptable in the study of chemical reactions. In this field what would

be great is the binding-energy errors being of the order 0.05 eV or less. The

simplicity of DFT calculational methods as compared to traditional many-

body techniques has, however, spawned considerable efforts to improve on

33

the LDA. Attempts to go beyond the LDA are based either on an improved

description of the exchange-correlation hole in real space or on a description of

exchange-correlation energies in reciprocal space usually leading to so-called

generalized gradient approximations (GGA). In recent years, the largest effort

has gone into the reciprocal-space approach which so far has been the most

successful ab initio DFT method. It is common and intuitive to consider

GGA corrections as some sort of next order corrections to the LDA. In this

framework a functional dependence on the gradient of the density is added

to ǫxc, i.e.,

EGGAxc [ρ] =

∫d3r ρ(r)ǫxc (ρ(r),∇ρ(r)) dr . (2.34)

As reported in [18], compared to the LDA approximation, the error for ion-

Figure 2.3: The errors (in eV) in the binding energies of the first-row dimersas obtained from different density functionals defined below. ∆ is the averageabsolute error for each functional.

ization energies is reduced by factors of 3-5 for the GGA corrections. In the

DFT community one can find several different GGA functionals. An illus-

trative comparison is made in [24]. The table in Figure 2.3 from the article

of Von Barth[29] gives a comparative idea of the accuracy of the corrective

functionals as well. Perdew, Burke, and Ernzerhof[25] have described a func-

tional form (PBE) that has several attractive features. The accuracy of the

34

PBE functional for atoms and molecules has been compared with results

of other popular functionals like LSD, BLYP, and B3LYP[40] by Ernzerhof

and Scuseria[26]. The PBE functional performed as well as B3LYP for the

properties considered by these authors. Generalized gradient approximations

generally lead to improved bond angles, lengths, and energies. In particular,

the strengths of hydrogen bonds and other weak bonds between closed shell

systems are significantly better than local density results. However, the self

interaction problem remains, and some asymptotic requirements for isolated

atoms are not satisfied.

2.4.4 Meta-GGA

Following the development of GGA, the next steps in the gradient approx-

imations were taken with a view to incorporate the kinetic energy density.

A version based on the PBE form was described by Perdew et.al.[28]. This

form includes the kinetic energy density for the occupied Kohn Sham or-

bitals. However, this and other forms initially developed included parame-

ter(s) found by fitting to experimental data. This last feature was avoided in

the recent work of Tao, Perdew, Staroverov and Scuseria[37][36] whose form

satisfied the requirement that the exchange potential be finite at the nucleus

for ground state one and two electron densities. Extensive numerical tests

for atoms, molecules, solids, and jellium surfaces showed generally very good

results.

2.4.5 Hybrid Schemes : Combination of Hartree-Fock

and DFT

In many reports it is mentioned just how poor the exchange energy differ-

ences could be between states whose wave functions have different nodal

structures. It has also been noted for many years[38] that errors in the local

density descriptions of exchange and correlation tend to balance. Probably

the final step in constructing the correct functional is to come up with the

Hybrid Functional. Hybrid functionals are a section of approximations to the

35

functional in DFT that blend a portion of Exact Exchange from Hartree-Fock

(HF) theory with exchange and correlation from other sources (ab initio, such

as LDA, or empirical). This suggests that a combination of Hartree-Fock ex-

change and DFT calculations could be useful as in the following expression,

EhybridXC = αEHF

X + EC , (2.35)

where α can be chosen to satisfy the particular criteria. A formal justifi-

cation for such hybrid schemes was given by Gorling and Levy[39]. Among

the ones which are used a lot these days in the community B3LYP[40] is

one. Hybridization with HF provides a simple scheme for improving many

molecular properties, such as atomization energies, bond lengths and vibra-

tion frequencies. It is to be noted here that those calculations which are done

with the first generation functionals enjoy a further improvement in accuracy.

Enrique R. Batista[41] and his co-workers made a benchmark calculations as

to have a comparative idea about the performance of the exchange correla-

tion functionals. According to their report the Diffusion Monte Carlo (DMC)

methods give the highest accuracy in terms of the calculations of interest.

Hence their comparing standard for all the exchange-correlation functionals

is the set of results from DMC. It goes without saying that DMC is very

expensive as far as computation time is concerned. It is an attempt to sum-

marize the comparative performances of the hybrid pseudopotentials. Ac-

cording to their report[41] the Heyd-Scuseria-Ernzerhof [42](HSE) functional

has been found to be in excellent agreement with their DMC benchmark (Fig-

ure 2.4). A comparative idea has been put into paper between the various

hybrid functionals like Perdew-Wang-91(PW91)[27], Tao-Perdew-Scuseria-

Staroverov [37](TPSS) and the previously mentioned functionals. The com-

parative idea shows that the hybrid functional like HSE significantly improves

the agreement between DFT and DMC (DMC being established as the most

accurate of all). This part of the research is still pretty open and further

works are still going on in trying to figure out the correct functional. As for

concluding remarks we can have a look at figure 2.5 where the Jacob’s ladder

is given describing the order of accuaracy for different functionals in practice.

36

Figure 2.4: (a)Difference in energy per atom in the diamond phase and inβ tin phase of Si. The HSE functional agrees with the DMC results whilethe other functionals underestimate the energy difference (b)Formation en-ergy of the three lowest energy single interstitials in silicon (X, H, andT).Comparison to the DMC results demonstrate a steady improvement ofthe accuracy of the functionals as the order of density expansion increases,with quantitative agreement for the hybrid HSE functional.[41]

The development of approximations to the exchange-correlation functionals

over the past 20 years has improved the performance of DFT calculations,

and many scientists in the community think that progress up the Jacob’s

ladder will continue until energy differences can be determined to within 1

kcal/mol (“chemical accuracy”). Of course the numerical cost increases as

one climbs, and this may not necessarily bring more information.

2.5 Pseudopotentials

Pseudopotentials were originally introduced to simplify electronic structure

calculations by eliminating the need to include atomic core states and the

strong potentials responsible for binding them. Most physical and chemi-

cal properties of crystals depend to a very good approximation only on the

37

Figure 2.5: Jacob’s ladder of DFT schemes according to Perdew and collab-orators.

distribution of the valence electrons. The core electrons do not participate

in the chemical bond. They are strongly localized around the nucleus, and

their wave functions overlap only very little with the core electron wave func-

tions from neighboring atoms. Hence, the distribution of the core electrons

basically does not change when the atoms are placed in a different chemical

environment. It is thus justified to assume the core electrons to be “frozen”

and to keep the core electron distribution of the isolated atom in the crystal

environment. This is essentially a non-technical description of the otherwise

technical approach known as Frozen Core approximation. The first advan-

tage of the frozen–core approximation is that now less electrons have to be

treated and less eigenstates of the Kohn–Sham equations have to be calcu-

lated. Secondly the total energy scale is largely reduced when the core elec-

trons are removed from the calculation which makes the calculation of energy

differences between atomic configurations numerically much more stable and

tractable. Figure 2.6 is the schematic illustration of an atomic all–electron

wave function and the corresponding atomic pseudo wave function together

with the respective external Coulomb potential and pseudopotential. ΦPS

and VPS are the pseudo wave function and the pseudopotential, where as the

solid line is ΦAE, the actual wave function. In the scope of the present thesis

38

Figure 2.6: Schematic illustration of an atomic all–electron wave function(solid line) and the corresponding atomic pseudo wave function (dashed line)together with the respective external Coulomb potential and pseudopotential.

let me discuss some of the important classes of pseudopotentials. depending

on the properties they exhibit they can classified and used in various schemes.

2.5.1 Normconserving Pseudopotentials

The pseudopotentials we use today are constructed from ab initio calcula-

tions for isolated atoms. The Kohn-Sham equations for a single atom are to

be solved for a single atom of the chemical species for which we would like

to generate a pseudopotential. This can be done without much of a prob-

lem since due to the spherical symmetry of atoms the wave functions can

be written as a product of a radial function and a spherical harmonic. The

Schroedinger equation then reduces to one dimensional differential equations

for the radial functions which can be integrated numerically. Figure 2.6 rep-

resents such a typical result for a radial function from such an “all–electron”

atom calculation together with the corresponding external Coulomb poten-

tial. The aim now is to replace the effective all–electron potential within a

given sphere with radius Rcut by a much weaker new potential with a node

less ground state wave function to the same energy eigenvalue as the origi-

nal all–electron state which matches exactly the all–electron wave function

39

outside Rcut. An important aspect here is to understand why should this

happen at all? We can notice that the radial Schroedinger equation for a

constant potential and fixed energy E to the angular momentum l is a one-

dimensional ordinary linear second order differential equation which has two

linear independent solutions. But only one of the two solutions, Φl(r), is

regular for r → 0. We have to see that if we do not change the logarithmic

derivative

Ll(E) =d

drlnΦl(r;E)|Rcut

=Φ′

l(Rcut;E)

Φl(Rcut;E)(2.36)

while changing the potential inside the atomic sphere, then the wave func-

tions outside the sphere remain unchanged. For the energy of the eigen-

state of our all electron calculation EAEl the procedure is as explained in the

following section. The all electron wave function ΦAEl inside the sphere is

replaced by an arbitrary smooth nodeless function ΦPSl with the equivalent

logarithmic derivative at Rcut as the original all electron function. Since ΦPSl

is nodeless by construction the radial Schroedinger equation can be readily

inverted with this new function and with the eigenvalue EAEl of the all elec-

tron calculation to get the potential that has exactly the required property

for the more complicated case at hand. A lot of recipies are in publication

with which this can be implemented[44][45][46][47][48][49][51]. One further

important requirement is the essential normconserving condition. In sim-

ple words this means that the all electron and the pseudo wave function

inside the atomic sphere must have the same norm to guarantee that both

wave functions generate identical electron densities in the outside region. In

addition to this condition, the additional degrees of freedom in generating a

suitable pseudopotential can be employed to make the pseudo wave functions

as smooth as possible[52]. Evidently till now the above arguments claim the

existence of the logarithmic derivative of the effective all electron potential

only for the reference energy EAEl . Now however, if the chemical environment

of the atom in consideration is changed, evidently the eigenstates will be at a

slightly different energy. Hence of course, for a pseudopotential to be useful it

has to be able to reproduce the logarithmic derivative of the all electron po-

tential over a whole energy range. The transferability of the pseudopotential

40

to other chemical environments depends on this width of the energy range.

As reported in various articles, in particular the normconserving condition

guarantees such a transferability. And again, the pseudopotential should be

as soft (technical in sense) as possible. The term soft means that the number

of plane waves required to expand the pseudo wave functions should be as

small as possible. Both properties, transferability and softness, are closely

related to the cutoff radius Rcut, and compete with each other. Reports claim

that low cutoffs give pseudopotentials with a very good transferability[50].

However, increasing Rcut makes the pseudopotentials softer. Usually com-

promising balance between the two requirements is to be struck with. An

upper limit for Rcut, is given by half the distance to the next nearest atom in

the configuration for which one may want to apply the pseudopotential. If

this value is exceeded, we can see that there will not be any region between

the neighboring atoms left where we will recover the true all electron wave

functions. Hence, we can not expect anymore to get an accurate description

of the chemical bond between the two atoms.

2.5.2 Fully Nonlocal Pseudopotentials

From (2.36) it is evident that the logarithmic derivative depends on the

angular momentum l and hence a separate pseudopotential V PSl (r) for each

value of l needs to be constructed. This would call for the fact that, the full

pseudopotential for our atom therefore has to be a nonlocal operator. The

following equation describes the way in which it is done.

VPS = V PSloc (r) +

∑

l

V PSnl,l (r)Pl, V PS

nl,l (r) = V PSl (r)− V PS

loc (r). (2.37)

The pseudopotential V PSl (r) pertaining to one specific angular momentum

(usually the highest value of for which a pseudopotential has been generated)

is taken to be the local part of the pseudopotential. The nonlocal compo-

nents V PSnl,l (r) are defined as the difference between the original l dependent

V PSl (r) and this local part of the pseudopotential. As we know by now that all

V PSl (r) are identical beyond Rcut, hence we understand here that the nonlo-

41

cal components of the pseudopotential are strictly confined within Rcut. Pl is

a projection operator which picks out the lth angular momentum component

from the subsequent wave function. This construction guarantees that when

the full pseudopotential operator VPS is applied to a general wave function

each angular momentum component of the wave function experiences only its

corresponding part V PSl (r) of the potential. Since the projection operators

act only on the angular variables of the position vector r the pseudopoten-

tial is still a local operator with respect to the radius . The form (2.37)

is therefore called a semilocal pseudopotential. As reported, for numerical

efficiency however, it would be better to have the pseudopotential in a fully

nonlocal form as described in the article by Kleinman and Bylander[53]. In

the following section the technique as shown by Vanderbilt[54] to construct

a fully nonlocal pseudopotential, is discussed.

2.5.3 Vanderbilt Ultrasoft Pseudopotentials

Figure 2.7: Illustration of a strongly localized valence wave function insidethe atomic core region and the modified wave function in the Vanderbiltultrasoft pseudopotential scheme.

The general view is that it is very difficult to treat within a pseudopo-

tential scheme all elements with nodeless valence states (in particular those

with 2p and 3s valence electrons). For those elements the pseudo and the all

electron wave functions are almost identical. Since these valence electrons are

strongly localized in the ionic core region, many plane waves are required for

42

a good representation of their wave function which often makes calculations

for such elements highly expensive. In consequent works[54][55] Vanderbilt

has introduced a new type of pseudopotentials the so-called ultrasoft pseu-

dopotentials, in which the normconserving requirement has been relaxed.

Instead of representing the full valence wave function by plane waves, only a

small portion of the wave function is calculated within the Vanderbilt ultra-

soft pseudopotential scheme (as shown in figure 2.7). This allows to reduce

substantially the plane wave cutoff energy in the calculations. However there

is a price to pay in terms of complications in calculations. It is because now

the Fourier representation of the Kohn Sham equation becomes more compli-

cated. First, when the electron density is calculated the part of the electron

distribution has to be added back (which is represented by the difference be-

tween the solid and the dashed line in figure 2.7, the so-called augmentation

charges). Secondly, due to the relaxation of the normconserving condition,

the Bloch eigenstates will not be not orthonormal anymore. An overlap

matrix has to be introduced and the eigenvalue problem of the Fourier rep-

resentation of the Kohn Sham equations will transform into a generalized

eigenvalue equation. Next, the nonlocal part of the pseudopotential becomes

density–dependent now. Lastly ,due to these modification additional terms

in the force calculations have to be evaluated. However, the report claims

that the gain in computational cost by lowering the plane wave cutoff energy

outweigh in many cases the additional computational effort which is required

by these modifications.

With all those theoretical concepts discussed it is now time to talk about

the present day theoretical models with which people are trying to tackle

the probelm of charged defects in solids. It is also important to know the

experimental techniques involved in the study of point defects. With that

view we move on to the next chapter.

43

Chapitre 3

Dans ce chapitre, les differentes techniques utilisees pour traiter les

defauts charges dans les solides sont abordees en details. Je commence avec

les techniques experimentales deja utilisees dans le silicium depuis plusieurs

annees, et detaille les approches utilisees pour chaque type de defauts. En-

suite, j’aborde les differentes methodes theoriques. Les differences dans

le traitement de l’electrostatique sont rapportees et les avantages et in-

convenients des differentes methodes sont prises en consideration, montrant

la motivation de d’avoir une autre methode. Parmi les methodes decrites, on

trouvera celle de Makov-Payne, celle de Schultz, celle de Freysoldt ou celle

de Dabo.

44

Chapter 3

State of the Art: Experiments

and Theory

3.1 Introduction

The success of the Density Functional Theory (DFT) as a tool for ab ini-

tio calculation of various properties of solids has inspired scientists to use

it even to study defects (both charged and uncharged) in metals and semi-

conductors. In this chapter I wish to give a moderate review of the most

widely used and successful techniques in calculating the potential and con-

sequently the energies and forces of systems with charged defects. To model

a localised defect nested in the bulk host, the supercell approximation is

employed by most of the calculations. Reports of increasing usage of this

method to predict the structures of crytalline solids and even an extension

to the aperiodic systems of defects and molecules are found since a long time

now [60][61][62]. According to the available reports it is easier to deal with

the uncharged point defects in metals and semiconductors with this method

than to deal with defects in ionic insulators. In reality all the approximations

that goes into the calculations viz. the used pseudopotentials, the choice of

exchange-correlation functionals or the shape and size of the supercell, can

account for a difference in the predicted physical properties. Among the few

important causes for this trouble at hand are the defect defect interaction

45

due to localised charge and the self interaction of the strongly localised de-

fect states. Hence it is arguably evident that the formation energies thus

inferred through this periodic techniques may not find a strong ground in

terms of theoretical completeness and comparison with the experimental re-

sults. In this chaper I would discuss in brief the various modifications in the

calculations undertaken so as to find the correct way to calculate the forma-

tion energies of the charged defects in bulk systems. It is notworthy here to

mention that broadly we can tackle the problem at two different points in

the run of calculations. We can either start off with fixing the potential of

the system with physical arguments and consequently generate an accurate

density for the self-consistent loop in the DFT calculations or we can incor-

porate correction terms to the finally calculated total energies. It is worth

mentioning at this point that the total energies that are calculated from an

inaccurate and approximate potential are bound to deviate from the true

values (experimental in all physical cases). We will have an overview in this

chapter of all the formalisms and derivations of various correction factors.

3.2 Experimental Methods To Study Point

Defects

Before taking a plunge into the various theoretical methods of modern day

to handle the charged point defects it can be a interesting as well to know

few important experimental techniques that are used in abundance to study

point defects. The most popular set of experiments which where done to

study the defects in silicon were done by Corbett and Watkins[3][63]. Those

experiments were carried out with the methods known as Electron Param-

agnetic Resonance (EPR) or Electron Spin Resonance (ESR). The theory

of such experiments is based on the fact where the atoms of a solid exhibit

paramagnetism as a result of unpaired electron spins. Here transitions can be

induced between spin states by applying a magnetic field and then supplying

electromagnetic energy, usually in the microwave range of frequencies. The

resulting absorption spectra are described as electron spin resonance (ESR)

46

or electron paramagnetic resonance (EPR). ESR has also been used as an

investigative tool for the study of radicals formed in solid materials, since

the radicals typically produce an unpaired spin on the atom from which an

electron is removed. The study of the ESR spectra has been very effective

for the radicals produced as radiation damage from ionizing radiation. Study

of the radicals produced by such radiation gives information about the loca-

tions and mechanisms of radiation damage. The interaction of an external

Figure 3.1: Schematic diagram of the theory behind the ESR experiment.

magnetic field with an electron spin depends upon the magnetic moment as-

sociated with the spin, and the nature of an isolated electron spin is such that

two and only two orientations are possible. The application of the magnetic

field then provides a magnetic potential energy which splits the spin states

by an amount proportional to the magnetic field which is nothing but the

Zeeman effect. Then radio frequency radiation of the appropriate frequency

can cause a transition from one spin state to the other. The energy associ-

ated with the transition is expressed in terms of the applied magnetic field

B, the electron spin g-factor g, and the constant µB which is called the Bohr

magneton. In their work Corbett and Watkins studied the EPR spectrum of

the silicon vacancies and reported the geometry, energy of formation and sta-

bility of the different types of point defects (vacancies). Another technique

which they used was the Electron-Nuclear Double Resonance (ENDOR). EN-

DOR is a magnetic resonance spectroscopic technique for the determination

of hyperfine interactions between electrons and nuclear spins. There are two

principal techniques. In continuous-wave ENDOR the intensity of an electron

47

paramagnetic resonance signal, partially saturated with microwave power, is

measured as radiofrequency is applied. In pulsed ENDOR the radiofrequency

is applied as pulses and the EPR signal is detected as a spin-echo. In each

case an enhancement of the EPR signal is observed when the radiofrequency

is in resonance with the coupled nuclei. George Feher[64] introduced this

technique in 1956 so that interactions which are not accessible in the EPR

spectrum could be resolved.

A method to calculate the formation energies of the point defects is to directly

calculate the type and concentration of the defects. This was put forward

by the works of R. O. Simmons and R. W. Balluffi[65]. Here it is necessary

to measure the change of the lattice constant a, i.e. ∆a, and the change in

the specimen dimension, ∆l, (one dimension is sufficient) simultaneously as a

function of temperature. Which in other words is also called the differential

thermal expansion method. Here l is the length of the specimen which can be

thought as a function of the temperature and the defects i.e. l(T, defects).

The basic idea is that ∆l/l–∆a/a contains the regular thermal expansion

and the dimensional change from point defects, especially vacancies. This

is so because for every vacancy in the crystal an atom must be added at

the surface; the total volume of the vacancies must be compensated by an

approximately equal additional volume and therefore an additional ∆l. If we

subtract the regular thermal expansion, which is simply given by the change

in lattice parameter, whatever is left can only be caused by point defects.

The difference then gives directly the vacancy concentration. Some of the

other direct methods for measuring point defect properties can be classified

as the following.

• There can be measurements of the resistivity. This method is pretty

suitable to ionic crystals if the mechanism of conduction is ionic trans-

port via point defects. But one may never know for sure if what is

being measured is actually the intrinsic equilibrium because ”doping”

by impurities may have occurred.

• Measuring specific heat as a function of T is another method. While

there should be some dependence on the concentration of point defects,

48

it is experimentally very difficult to handle with the required accuracy.

• Measuring electronic noise can also be a direct method in this category.

This is a relatively new method which relies on very sophisticated noise

measurements. It is more suited for measuring diffusion properties, but

might be used for equilibrium conditions, too.

All the methods described above are for the equilibrium point defects. One

must undertake Quenching Experiments for the non-equilibrium point de-

fects. The basic idea behind these techniques is simple. If one has more

point defects than what one would have in thermal equilibrium, it should

be easier to detect them. There are several methods, the most important

one being quenching from high temperatures. The general methodology is

as follows.

• A wire of the material to be investigated is heated to some desired

(high) temperature T in liquid and superfluid He II (i.e. a liquid with

somewhat large heat conduction) to the desired temperature (by pass-

ing current through it). Surprisingly this is easily possible because the

He-vapor produced acts as a very efficient thermal shield and keeps the

liquid He from exploding because too much heat is transferred.

• After turning off the heating current, the specimen will cool extremely

fast to He II temperature (> 1K). There is not much time for the point

defects being present at the high temperature in thermal equilibrium to

disappear via diffusion; they are, for all practical purposes, frozen-in.

The frozen-in concentration can now be determined by e.g. measuring

the residual resistivity of the wire.

• The residual resistivity is simply the resistivity found around 0 K. It

is essentially dominated by defects because scattering of electrons at

phonons is negligible.

A relatively new way of looking at point defects is to use the scanning

tunneling microscope (STM) and to look at the atoms on the surface of the

sample. This idea is not new actually. Before the advent of the STM field ion

49

microscopy was used with the same intention, but experiments were (and are)

very difficult to do and severely limited. One idea is to investigate the surface

after fracturing the quenched sample in-situ under ultra-high vacuum (UHV)

conditions. This would give the density of vacancies on the fracture plane

from which the bulk value could be deduced. Although in these experiments

the vacancies can be seen, but there are other problems to tackle with. The

image changes with time - the density of point defects goes up.

In a nutshell I have tried to give an overview of the experimental techniques

that are involved in detecting the various properties of defects in the solids.

From the next section we would concentrate only on the theoretical models

to study defects in general.

3.3 Theoretical Methods To Study Point De-

fects

3.3.1 Treatment of Makov and Payne

In their article G.Makov and M.C.Payne[66] explained in length the tech-

nique of using Periodic Boundary Conditions (PBC) for different cases of

practical theoretical interest. They have argued the validity of using PBC

for calculations in solids on grounds like (i) ease of implementation in the cal-

culation (ii) compatibility with plane-wave expansions and consequent simple

calculations of forces in molecular dynamics simulations. (iii) availability of

an unified scheme for both periodic and aperiodic systems. In the article

they have proposed a detailed description of the electrostatics of the system

dealt with. The need for the correct electrostatics is felt as the convergence of

the total energy, with the increase of the size of the supercell, is determined

by the longest-ranged forces. Those forces are indeed electrostatic in nature.

The elastic forces can also be long ranged in solids and they are also treated

in the similar fashion in this particular report. In principle this work is a

continuation of the works by Leeuw[67] et al. The idea is to calculate the

total energy as correctly as possible because all the other physical quantities

50

are derived from the energy via the variational principle. For aperiodic sys-

tems using PBC, one is only interested in the energy, E0 in the limit L → ∞,

where L is the linear dimension of the supercell. The energy calculated for a

finite supercell E(L) differs from E0, because of the spurious interactions of

the aperiodic charge density with its images in the neighbouring cells. The

first case to investigate is the case with Neutral molecular species.

Neutral molecular species

Here for simplicity it is considered that the entire molecular charge density

is contained in the supercell. In practice, the electron density decays ex-

ponentially away from the molecular species and the supercell need only be

large enough for the density at the surface of the cell to be in this regime.

As reported the energy is absolutely convergent for the isolated molecule

which has no dipole moment. A careful conjecture about the nature of the

asymptotic behaviour of the energy would show that the energy will be,

E(L) = E0 +O(L−5). (3.1)

For an isolated neutral molecule that has a dipole moment the scenario is of

course different as the order of summation of the electrostatic sum should be

chosen with a bit more care. For an infinite lattice as considered in this type

of formalism that sum is not well defined. Hence the choice of the summation

area should be more of convenience with respect to the extent of the dipole

moment. In this case of course one finds the presence of a dipole dependent

term. It is also to be noted that for an aperiodic system, the absence of

this term will not change the value of energy in the limit L → ∞, as the

extra term is O(L−3). The results (figure 3.2) of a calculation from their

article show that energy calculated without the dipole term converges slowly

in comparison to the one, done with the dipole term. Another interesting

observation can be mentioned here. In general the dipole moment is reported

to be ill defined in PBC[68]. In principle, in a periodic solid, all choices of

supercell geometry relative to the charge distribution should give the same

energy. The electrostatic energy functional as menthioned in [66] infers that

51

Figure 3.2: The energy calculated without the dipole term converges slowlyin comparison to the one, done with the dipole term

different choice of the supercell will give different values of the dipole moment.

This is the manifestation of the fact that an infinite periodic solid with a

dipolar basis will not have a well defined energy. The limit L → ∞ dictates

the choice of the necessary supercell geometry. The supercell then must

create the picture of the aperiodicity in the cell as it would be in the bulk.

This is indeed necessary for the dipole moment to be invariant. For the case

considered in [66] the shape of the supercell is chosen to be a cube as the

result implies that dipoles on a cubic lattice do not interact and hence the

convergence of the energy would be O(L−5).

Nonneutral molecular species

Systems dealing with charged impurities or vacancies in crystalline solids

would call for total energy calculations in a periodically repeated electrically

charged system. And of course the value of the total electrostatic energy

diverges. In this scenario what is considered is the original charged system

52

immersed in a Jellium background, so the net charge is zero[62]. Then again

one can fall back to the procedure undertaken previously for the neutral

molecular species. The interaction between the Jellium background and the

charged species will decrease with the cell size and hence the convergence

will be slow, having a form of a power law in L. Evidently the charge density

of the immersed system consists of the density of the charged species and

Jellium density. This density again can be thought of as a contribution of

two charge densities giving rise to self interactions and an interaction with

each other. If we consider a charge placed at a particular point inside the

supercell, the self interaction of that charge will give rise to the term more

well-known as the Madelung energy of a lattice of point charges immersed in

a neutralizing Jellium.

EMadelung = −q2α

2L, (3.2)

where α is the lattice dependent Madelung constant. The mutual interaction

of the charge densities will also give rise to a very important term. As the

Jellium density depends on the supercell size hence this energy does not

converge as the previous one. The form includes a quadrupole term signifying

the interaction between the neutral defects. At the end what we get are two

correction factors. The assymptotic result for the total electrostatic energy

of a charged species on a cubic lattice finds the form as

E = E0 −q2α

2L− 2πqQ

3L3+O(L−5), (3.3)

where Q is the quadrupole moment and E0 is the desired electrostatic energy

of the isolated species. Results with ionization potential of a Mg atom are

shown in Figure 3.3. With all the corrections applied, it is evident that the

convergence is faster.

Aperiodic system in condensed matter

For aperiodic system in condensed matter the electrostatic energy can be

considered to be the sum of three major interactions. (1) The periodic charge

density interacting with itself. This is independent of L. (2) Interaction

53

Figure 3.3: (a)The ionization potential of a Mg atom calculated in cubicsupercells of side L (triangles), The same after including Madelung correction(squares), The same after the entire correction factor is applied (circles) (b)Expanded section of (a)

between periodic and aperiodic charge densities. This is also independent

of L.(3) The L dependent aperiodic densities located in different supercells.

Here also the main motive is to chalk out the asymptotic dependence of the

aperiodic density and its multipoles on the supercell dimensions. The first

source of L dependence is the one already discussed above, the case of changes

in the charge distribution induced by the interactions of the aperiodic charge

with its images. The second reason for the L dependence is the dielectric

response of the periodic medium to the aperiodic density. This is treated

nonelectrostatically and a correction factor containing the dilectric constant

is incorporated in the spirit of the arguments of Leslie and Gillan. The term

contains a quadrupole moment which is an artifact of the aperiodic density

whereas the dielectric constant is coming from the periodic density. With

this correction the final form of the total electrostatic energy becomes

E = E0 −q2α

2Lǫ− 2πqQ

3L3ǫ+O(L−5). (3.4)

In this case though, Q is the second radial moment only of that part of the

aperiodic density that does not arise from the dielectric response or from the

the Jellium. It is worthy of mention here that both Q and ǫ are properties

of the aperiodic density and the periodic density respectively.

54

3.3.2 Treatment of Schultz

A careful analysis by Peter A. Schultz[69] deals with some of the problems in

the Makov Payne formulation and employs a different method to avoid them.

Schultz concludes in his analysis that the Jellium approach fails to represent

the topology of the local electrostatic potential accurately. In fact he men-

tions in his article that the inaccuracy is to such an extent that for typical

semiconductors with sufficient band-gap energies the results obtained in the

Jellium approach are delusive. It is reported that tails of the potentials gen-

erated by local electrostatic moments of the defect overlap cell boundaries,

corrupting the local potential of the “isolated” defect. For supercells sep-

arated by vacuum gaps, e.g. molecules or repeated slabs, it is possible to

avoid, rigorously, the error of interacting multipole moments. He proposes a

technique as a generalisation of the Local Moment Counter Charge (LMCC)

method [70] to calculate a more accurate electrostatic potential within the

supercell. In this article the error in the predicted potential of the Jellium

approach in comparison to his LMCC method is also reported. The main

problem is that the local potential incorporates an error with the potential

tails of the periodic image charges which the compensating Jellium charge

does not take into account. The results show that the error in the potential

for semiconductors like charged Si and Ge are in the order of their band-gaps

viz. 1.00 eV for Si 64 atom cubic supercell and 0.62 eV for Ge 216 atom

cubic supercell. Increasing the cell size to reduce the Jellium potential error

is considered impractical in this article. In Figure 3.4 the error in the com-

puted electrostatic potential in a supercell calculation using jellium as the

neutralizing agent has been shown. As reported for vacuum gap supercells,

the LMCC method[70] is an alternate solution to Poisson’s equation that ex-

actly removes the spurious effects on the potential (and energy) of multipole

moments in the supercell. According to this formulation of Schultz, a model

density composed of a Gaussian array of charges is constructed in conformity

to the moments of the local system. Hence the system’s total charge distri-

bution comprises two parts, one part from this model density, and another

remaining half, which is devoid of any moments. Hence the potential for this

55

Figure 3.4: Error in the computed electrostatic potential in a supercell cal-culation using jellium as the neutralizing agent. The exact potential is com-puted analytically for a Gaussian density distribution in a cube, and usingan FFT after neutralization by jellium. The difference is plotted along a lineconnecting the centers of opposite faces and going through the center of thecell (inset).

part can easily be calculated with sufficient accuracy employing PBC and

Fast Fourier Transformation (FFT). The model density is treated as isolated

and only then the potential is calculated. Evidently the total potential is

the sum of these two, viz. the PBC potential and the Local Moment(LM)

potential,

φ(r) = φ′

PBC + φLM (3.5)

The idea is to create an isolated defect or molecule and blending the bound-

ary conditions to achieve the solution to the Poisson’s equation. For charged

defects in Si the potential φLM is found to be discontinuous and this phe-

nomenon needs to be avoided. The qrlong range asymptotic potentials mis-

align at cell boundaries. Again if the LMCC charge cannot be fixed by sym-

metry, the multipole moments of the defect are to be exactly determined.

In reality these are ill defined in bulk. Lastly the interaction of the LMCC

charge with the bulk material is to be taken into account. The potential of

an isolated charged defect extends outside the local cell volume and interacts

with bulk, but the LMCC potential is truncated at the cell boundary. The

Wigner-Seitz cell geometry around the position of each LMCC charge takes

56

care of the discontinuity in the potential. The multipole moments of the

defect can be calculated in two halves, contributions coming from the sum of

the spherical(momentless) atom charge distributions and the cell-dependent

moments of the perfect crystal reference and the defects. The last part of this

technique involves the interaction of the local charge with the bulk medium.

Schultz shows two Madelung integrals accounting respectively for (i)the in-

teraction between the LMCC and the potential from the crystal density and

(ii) the interaction between the LMCC potential and the crystal density in

WS volume.

EM =

∫nLM(r)δφref (r)−

∫

WS

φLM(r)δρref (r). (3.6)

The first term is a Madelung integral over all space between the LMCC

and the potential δφref (r) from the crystal density δρref (r) The second term

subtracts a local Madelung integral of the LMCC potential and a crystal

density in the WS volume. The energy of the dielectric response of the bulk

is calculated in same spirit as in the approach of Leslie and Gillan[71]. The

claim in this method is that it eliminates the spurious interaction of charges

between supercells and helps to compute the properties of charged defects

in bulk systems with the help of correctly computed electrostatic potential.

Results from three series of calculations are presented in the paper (figure

3.5): a 1D single strand of alternating Na and Cl, a 2D square single sheet

of NaCl, and 3D bulk NaCl. Also from Figure 3.4 it is to be noted that

the resulting energy error in the jellium-based calculation is not linear in

the charge. Doubling the charge doubles the error in the computed poten-

tial, and the energy is the integral of these two. Hence, the error in total

energy calculations scales at least as q2 , even neglecting the consequences

of screening. If results for singly charged systems bear uncertainties on the

order of a band-gap energy, results for multiply charged defects will be much

worse. In this study though the first principle calculations of total energetics

of charged vacancies in NaCl that accurately embody the correct electrostatic

interactions appropriate to isolated defects is reported with some criticism

of the Makov Payne formalism. It is demonstrated that the standard recipe

57

Figure 3.5: (a) The computed IP for a chlorine vacancy in a 1D NaCl chain,as a function of cell size. (b) The computed ionization potential for a chlorinevacancy in a 2D square NaCl sheet, as a function of cell size. Points frompolar NXN cells (inset: smaller 1X1 two atom cell) are plotted with squares,and nonpolarN

√2XN

√2 cells (inset: larger rotated four-atom

√2X

√2 cell)

with diamonds. (c) The computed ionization potential for a chlorine vacancyin bulk NaCl, as a function of cell size. Points from nonpolar cubic cells areplotted with squares, and points from polar fcc cells with diamonds. Thedashed line denotes the extrapolated asymptotic limit of the IP.

for studying charged defects in bulk systems, using jellium as a neutralizing

agent in supercell calculations, incorporates an error in the computed elec-

trostatic potential comparable to the energy scale of physical interest. The

generalized LMCC approach presented here claims to eliminate the unphys-

ical interaction of charges between supercells and enables the computation

of the properties of charged defects in bulk systems using the correct local

electrostatic potential.

58

3.3.3 Treatment of Freysoldt, Neugebauer and Van de

Walle

In their report Freysoldt[72] et. al. argues against the formalisms of the

correction factors of Makov and Payne and even the potential correcting

schemes Schultz. They argue that for realistic defects in condensed sys-

tems, however, the quadrupole moment correction term as stated in previous

works [66] do not always improve the convergence[73][74][75]. They have

shown that the Madelung correction greatly overshoots for small supercells.

Alternatively they have also argued against the technique of Schultz type for-

mulation where several scientists have suggested to truncate or compensate

the long-range tail of the bare Coulomb potential during the computation

of the electrostatic potential itself in order to remove the unwanted interac-

tions. According to Freysoldt[72] et. al. this scheme when applied to solids

suffers from ignoring the polarization outside the supercell. In their approach

they put forth a new idea which they claim (i) is based on a single super-

cell calculation, (ii) does not rely on fitted parameters, (iii) derives from an

exact expression applying well-defined approximations, and (iv) sheds light

on the problems encountered in previous schemes. The central idea of this

type of method revolves around the accurate calculation of the potentials

and proper treatment of the defect-defect interactions. The creation of the

charged defects is broken into three building steps which would account for

the final expression of the energy. First the charge for a single defect is in-

troduced by adding or removing electrons. At the same time all the other

electrons are frozen. This step results in an unscreened charge density. Ob-

viously enough the following step would incorporate all the other electrons

to screen this introduced charge. Lastly an artificial periodicity is imposed

upon along with the addition of a compensating homogeneous background

charge. With all these inputs and considering the unscreened defect charge

to be fully contained in a convenient zone of the supercell the screened lattice

energy of the charge with the compensating background is formulated out.

Here one must take care that when the electrons are allowed to screen the

introduced charge, the electron distribution forces an alteration in the elec-

59

trostatic potential Vdefect with respect to the neutral defect, which is given

by,

Vq/0(r) = Vdefectq − Vdefect0 . (3.7)

The resulting potential then is (up to an additive constant) a superposition

of the potentials Vq/0(r + R), where R is a lattice vector. Now this Vq/0 in

the reciprocal space would be easily formulated with the Fourier transforma-

tion. Hence the periodic defect potential is obtained from a Fourier series as

Vq/0(r). The interaction energy associated with the artificial potential due

to the periodic repetition of the charge comes out to be

Einter =1

2

∫

Ω

d3r[qd(r) + n][Vq/0(r)− Vq/0(r)], (3.8)

where n is the homogeneous background charge −q/Ω (Ω being the volume

of the supercell), qd(r) is the unscreened charge density and Vq/0(r)−Vq/0(r)

is the artificial potential due to the periodic repetition. The prefactor 12

accounts for double counting and the integral is restricted to the supercell.

Now the energy arising from the interaction of the background charge with

the defect inside the reference cell is calculated out in the following way,

Eintra =

∫

Ω

d3rnVq/0(r) = −q(1

Ω

∫

Ω

d3rVq/0(r)). (3.9)

Here n = −qΩ, the compensating homogeneous background charge. Adding

and rearranging (3.8) and (3.9) would allow the authors to explicitly calculate

the artificial interactions from parameters like ǫ, qd, Vq/0, which can then be

subtracted from the uncorrected formation energies obtained from the ab

initio supercell calculations. Theoretical value of ǫ is chosen (which can be

computed from various theories). For qd , it turns out that any reasonable

approximation to the defect charge distribution suffices since the sum of

lattice energy and alignment correction is not sensitive to the details of qd.

And Vq/0 is available directly from the DFT supercell calculations. The final

form of the formation energy for this paper is written down as (results of

unrelaxed Ga vacancy for a set of 2 × 2 × 2 supercells of the 8-atom cell are

60

reported here),

Ef (V qGa) = E(V q

Ga + bulk)− E(bulk) + E(Ga) + µGa − Elatq + q(EF +∆q/b),

(3.10)

where the last two terms the above equation are obtained by rearrangement

of the sum of (3.8) and (3.9) as

Einter + Eintra = Elat − q∆q/0. (3.11)

The Fermi energy EF is set equal to the valence-band maximum and the Ga

chemical potential µGa to that of Ga metal. The results are shown in figure

3.6. Using a point-charge model qd the ab initio corrected formation energies

Figure 3.6: Defect formation energies of the Ga vacancy in GaAs includingsupercell corrections as a function of the inverse number of atoms. TheFermi energy is set equal to the valence-band maximum and the Ga chemicalpotential to that of Ga metal.

of all calculations above 2 X 2 X 2 agree within 0.1 eV without any empirical

fit.

3.3.4 Treatment of Dabo et.al.

In the works of I. Dabo, B.Kozinsky, N.E.Singh-Miller, and N.Marzari[76]

argues in favour of their analysis to compute the correct electrostatic poten-

61

tial for charged systems in the framework of periodic calculations. The paper

even deals with extensive comparison with other methods of similar compu-

tation. In this work, they propose an alternative approach for correcting pe-

riodic image errors and show that energy convergence with respect to cell size

can be obtained at moderately available computational costs. The approach

advances by considering the exact electrostatic potential in open-boundary

conditions or any other boundary condition as the sum of the periodic so-

lution to the Poisson equation computed using FFT techniques (which are

inexpensive in reality) and a real space correction that can be obtained on

coarse grained meshes with multigrid techniques within a chosen degree of

accuracy. Hence basically the idea revolves around the trick of generating

a corrective potential on the basis of the boundary conditions. The correc-

tive potential is a mutual contribution from (in fact the difference between)

the potential arising from the Open Boundary Conditions (OBC) in absence

of an external electric field and the potential (satisfying the PBC) due to

the periodic translation of the same charge distribution. The corrective po-

tential satisfies much simpler, smoother differential equation (essentially the

Poisson’s equation).

∇2vcorr(r) = −4π(ρ), (3.12)

where vcorr is the corrective potential and ρ is an isolated charge distri-

bution satisfying the Poisson equation. It may be said here that a careful

and logical manipulation of the electron density ρ gives the corrective poten-

tial vcorr. The adjoining figure 3.7 gives an idea of the potentials for carbon

monoxide absorbed in on a neutral platinum slab and carbon monoxide ad-

sorbed on a charged platinum slab. The corrective potential seems to vary

more smoothly than the potentials following OBC or PBC. This scheme is

also tested with a point charge in a periodically repeated cubic cell and it is

found that the corrective potential needs to be expanded in a power series

exploiting the symmetry of the cubic cell for the convenience of computa-

tion. In Figure 3.8 we see the corrective potential in the presence of a point

charge q = +e in a periodically repeated cell of length L. The parabolic

expansion (valid up to third order) confirms that the point charge correction

62

Figure 3.7: Open-boundary electrostatic potential v, periodic potential v′

and electrostatic-potential correction vcorr averaged in the xy plane parallelto the surface for (a) carbon monoxide adsorbed on a neutral platinum slaband (b) carbon monoxide adsorbed on a charged platinum slab.

Figure 3.8: Corrective potential vcorr for a cubic lattice of point charges andits parabolic approximation in the vicinity of the origin.

or in reality the corrective potential is almost quadratic around the origin of

the system. Strikingly mathematical analyses yield that it is even quadratic

up to the third order for non-cubic lattices. Final form of the electrostatic

63

correction vcorr for an arbitrary distribution ρ would look like,

vcorr(r) =

∫vcorr0 (r − r′)ρ(r′)dr′, (3.13)

where the corrective potential generated by the uniform jellium and the sur-

rounding point charges is denoted by vcorr0 . The authors have compared

this scheme with some of the existing schemes and found that the correc-

tive potential gives out the Madelung constant and even a term similar to

the Makov Payne formulation, but differing slightly in magnitude though.

Three different cases are compared in this approach. One with a Parabolic

Point-Counter charge (PCC) potential. This one claims that this scheme

can correct periodic image errors up to quadrupole moment order. To obtain

higher-order PCC corrections, one would need to determine more terms in the

expansion of the point-charge correction beyond the parabolic contributions.

Another popular choice is to implement the Gaussian Counter charge (GCC)

formalism where the corrective potential is Gaussian. In this approach the

corrective potential is claimed to easy in computation. The GCC method is

somewhat similar to methods applied by Blochl[80] and LMCC method of

Schultz. At this point the authors felt the need for another approach which

is centered on the idea of the direct difference between the open boundary

potential and the periodic counterpart. This exact corrective potential is

coined as the Density Counter charge (DCC) potential by the authors. The

DCC potential is obtained by evaluating the Coulomb integral defining v

at each grid point in the unit cell. The PCC, GCC, and DCC potentials

for a charged pyridazine cation in a cubic cell of length L = 15 bohr are

plotted in figure 3.9. The PCC and GCC corrections are computed up to

the dipole order. First, it should be noted that the maximal energy of the

PCC potential is slightly above its GCC counterpart, reflecting the fact that

the Madelung energy of an array of point charges immersed in a jellium is

higher than that of a jellium-neutralized array of Gaussian charges. In ad-

dition, the maximal DCC energy is found to be above the Madelung energy,

proving that the dipole PCC and GCC corrections tend to underestimate the

energy of the system. Moreover, the parabolic PCC potential is not as steep

64

as its GCC counterpart, suggesting that the energy underestimation will be

more significant for the GCC correction. Because of the cubic symmetry of

the cell, the PCC and GCC potentials display the same curvature in each

direction of space. In contrast, the curvature of the DCC potential is not

uniform due to the nonspherical nature of the molecular charge density. This

shape dependence suggests that the accuracy of the GCC correction could

be improved by optimizing the geometry of the Gaussian countercharges.

Figure 3.9: PCC, GCC, and DCC corrective potentials for a pyridazine cationC4H5N

+2 in a cubic cell of length L = 15 bohr. The corrective potentials are

plotted along the z axis perpendicular to the plane of the molecule. ThePCC and GCC corrections are calculated up to dipole order.

∆Ecorr =1

2

∫v(r)ρ(r)dr. (3.14)

Equation (3.12) gives the final form of the corrected energy pertaining to

the corrected potential.

In Figure 3.10 the total energies of the Pyridazine cation have been plotted

with cell size in the different corrective schemes. From the figure 3.10 it is

evident that with the DCC scheme, the energy converges even more rapidly,

reflecting the exponential disappearance of energy errors arising from the

charge density spilling across periodic cells. Further treatments to other sys-

tems in one and two dimensional periodicity are still being worked upon. This

65

Figure 3.10: Total energy as a function of cell size for a pyridazine cationwithout correction and corrected using the PCC, GCC, and DCC schemes.The PCC and GCC corrections are calculated up to quadrupole order. Theinset shows a pyridazine cation in a cell of size L = 15 bohr.

scheme in general has been reported to have improved upon the drawbacks

of the Makov Payne or Schultz type formalism.

3.4 Conclusion

All the above mentioned formalims argue in favour of the periodic calcu-

lations for the charged systems. The idea is to correct for the erronious

description of the electrostatics because of the periodicity of the supercell

approach. We wonder if this is the only way to compute the physical quan-

tities of interest for systems with charged defects. Each process comes with

certain advantages over the other one, yet the fact remains as truth that

electrostatics of the charged system is not correctly portrayed in the PBC

schemes. The convergence issues are also of prime importance. All the above

mentioned methodologies exploit the DFT methods to calculate the total

energies and potentials required for the respective studies. The places where

we wish to probe in more is the search for a Free Boundary Condition (FBC)

scheme (under the DFT framework) with the exact electrostatics of the sys-

tem. We wish to check the validity of the pseudopotential approach as well

with a comparative idea of all the exchange correlation functionals and the

66

corresponding pseudopotentials for our case study. Before taking the ride

through the new method of treating charged deefects I would talk through a

chapter where a norm-conserving family of pseudopotentials is tested for its

accuracy. To make it even more accurate non linear core correction has been

added and further tested. The next chapter would deal with such a topic.

67

Chapitre 4

Dans ce chapitre est aborde l’utilisation de la correction non-lineaire de cœur

avec des pseudo-potentiels HGH afin d’ameliorer la precision des calculs ef-

fectues avec pseudo-potentiels a norme conservee, justifiant ainsi l’utilisation

d’un formalisme plus simple compare a la methode des projecteurs aug-

mentes (PAW). La correction non-lineaire de cœur est une facon de corriger

la linearisation apportee par l’utilisation de pseudo-potentiels. Ce chapitre

montrera la validite d’une telle approche ainsi que les resultats obtenues sur

les energies d’atomisation.

68

Chapter 4

Hartwigsen-Goedeker-

Hutter(HGH) pseudopotentials

with Non Linear Core

Correction (NLCC)

4.1 Introduction

As been discussed in the previous chapters, the fact DFT has proved itself

to be an extremely efficient approach towards the description of ground state

properties of metals, semiconductors and insulators motivates us to use it for

our first stretch of calculations. In the formalism of DFT, we have seen the

importance of a approximation –the exchange correlation functional. Within

a given range of error bar for this approximation it is already established that

DFT is the most widely used method for the electronic structure calculation

in the fields of solid state physics and quatum chemistry, as reports have

have been abundant in the community that, in the framework of DFT hun-

dreds of atoms can be dealt with immense accuracy and efficiency. Another

important aspect of the calculations that one may do with DFT is the use of

pseudopotentials. The pseudopotentials are used to simplify the lives of re-

searchers as they simply the calculations with less computation cost and they

69

also gives us a way to avoid the Dirac equation. Other wise we would have to

use the Dirac equation to take take into account the relativistic effects of the

all electron picture. It is known from theoretical quantum mechanics that the

wavefunction ocillates near the core for any species because of the core elec-

trons strong interaction with nucleus. This makes the all electron potential

very complex to deal with in computational techniques. The pseudopotential

is an effective (unreal of course) potential constructed to replace the atomic

all-electron potential (Full-potential) such that core states are eliminated and

the valence electrons are described by nodeless pseudo-wavefunctions. In this

approach only the chemically active valence electrons are dealt with explic-

itly, while the core electrons are ’frozen’, being considered together with the

nuclei as rigid and non-polarizable. Norm-conserving pseudopotentials are

derived from an atomic reference state, requiring that the pseudo and all-

electron valence eigenstates have the same energies and amplitude (and thus

density) outside a chosen core cutoff radius rc. Pseudopotentials with larger

cutoff radius are said to be softer, that is more rapidly convergent, but at

the same time less transferable, that is less accurate to reproduce realistic

features in different environments. Norm-conserving pseudopotentials em-

phasizes on the condition that, outside the cutoff radius, the norm of each

pseudo-wavefunction be exactly same to its corresponding all-electron wave-

function. We have used Hartwigsen-Goedeker-Hutter HGH pseudopotential

for this part of our calculations. The main idea is to use the pseudopo-

tentials for accurate calculations. One can argue of course the validity of

such a job. The emphasis is on the complexity-less approach of the norm-

conserving pseudopotentials to produce as accurate results as that of the

Projector Augmented Wave[80] (PAW) approach or that of the all-electron

calculations. The success of our approach would ensure a method other than

the PAW method for accurate calculations of various physical quantities.

Here in this work another important concept is discussed, the Non Linear

Core Correction (NLCC). This is an attempt to correct the otherwise im-

plied linearization of the pseudopotential. So in a nutshell this chapter will

deal with the validity of the results of calculations done with the Hartwigsen-

Goedeker-Hutter(HGH) pseudopotentials with NLCC . Another important

70

point to be mentioned here is the fact that this pseudopotentials are imple-

mented with the PBE exchange-correlation functionals on the wavelet basis

sets. Along with the exchange-correlation functional, the representation of

the Kohn-Sham wave functions is a prime concern for the DFT calcula-

tions. Usage of system independent basis sets such as plane waves[80]-[82],

wavelets[43], or real space grids[83][59] a very high level of accuracy can be

achieved. Basis sets of these kinds can be expanded systematically to achieve

convergence according to our level of confidence. Among the systematic ba-

sis sets wavelets are reported to have all the properties that are needed for

an accurate description of the system at hand. Systematic basis sets like

these even allows us to implement correctly and more conveniently the hy-

brid functionals. As reported earlier[43], such a method is implemented in

a DFT code, viz. BigDFT. As mentioned earlier it can be further clarified

that the goal of this work is to show the results of the implementation of

PBE functionals on wavelet basis. For the test the G2-1[84] set of molecules

are taken and the atomization energies of these 55 molecules are calculated

within the framework of PBE functionals implemented in BigDFT. The re-

sults of this work is duly compared with the known experimental values and

also with values reported from the calculation from VASP and Gaussian03

DFT codes[85].

4.2 Theory

4.2.1 Non Linear Core Correction

The need for NLCC

In their report[87] Louie and his colleagues have put forward a method to

extend the pseudopotential study primarily for the magnetic systems. The

aim is to deal with problems concerning properties of magnetic materials,

spin-density waves, magnetic effects on surfaces, localized impurity states in

defects, etc., without much computational trouble. The method also takes

care in keeping the level of accuracy as it has been in the nonmagnetic cases.

Previous to their works, to take into account the magnetic effects into pseu-

71

dopotential calculations, in their article, Zunger[88] proposed constructing

separate ionic potentials for the spin-up and spin-down electrons. Essen-

tially they devised a spin-dependent pseudopotential approach. And in here

the ionic pseudopotential in the solid depends on the spin density of the va-

lence electrons. This spin density is obtained by interpolation between the

ionic potential for the paramagnetic atom and that of the fully spinpolarized

atom.

The work of Louie and his collaborators differs from the work of Zunger with

a view that it is unnecessary and often undesirable to employ these spin-

dependent ionic pseudopotentials. As it is well known from the literature

mentioned in the previous chapters the exchange and correlation energy is

usually approximated by some local (nonlinear) function of the charge den-

sity, and the kinetic energy is found from the gradient of the now obtainable

singleparticle wave functions. As we have also seen that the charge density,

in the pseudopotential formalism, is divided into core and valence contri-

butions, and the energy of the core is assumed to be constant. Since the

energy of the core is not varying it is hence subtracted out from the total

density. Also it is to be kept in mind that quite often the core contribution is

absolutely neglected, and the total energy is given by the DFT energy func-

tional with the total charge density replaced by a (pseudo) valence charge

density. The electrostatic potential due to the interaction of the ions with

the core electrons is replaced by the pseudopotential. It is now pretty clear

that with this trick all the interaction between the core and valence electrons

is thus transferred to the pseudopotential. From the above arguments we can

understand that neglecting the core electrons poses a problem. This mode

of treatment linearizes the interaction which in reality is an approximation

to the kinetic energy and as well to the explicitly nonlinear exchange and

correlation energy. More explicitely, it is assumed in this fashion.

EXC(ρc + ρv) = EXC(ρc) + EXC(ρv), (4.1)

where EXC(ρc + ρv) is the total exchange correlation energy (non linear in

reality). The form shows how it is linearized. At this point it is also im-

72

portant to look for the cases where this approximation might have worked

very nicely. There can be cases where the core and the valence charge den-

sities are well separated in space. For example we can consider the cases

of various nonmagnetic molecules. Quite naturally there will be no serious

errors introduced here. Most of the results reported till date hold good only

because of this apparent spatial separation of the core and valence charge

densities. But of cource where there is significant overlap between the two

densities, the linearization, in particular of the exchange and correlation, will

lead to the problem of transferability of the pseudopotential. The fact that

the transferability is reduced, in turn, would incorporate systematic errors

in the calculated total energy. To deal with the problem one must under-

stand the basis nature and the origin of it. In the spin-density formalism the

exchange and correlation energy depends on the local spin density as well as

on the charge density. Now this additional spin dependence introduces the

additional nonlinearity which is the main point of discussion in this mod-

ule. Moreover it is the errors introduced by the linearization described above

that have caused the discrepancies between the experimental and theoretical

calculations and hence it is now necessary to use spin-dependent ionic pseu-

dopotentials. The scheme that is to be used now will formally be addressed

as Non Linear Core Correction (NLCC). NLCC treats these nonlinear terms

explicitly, and the need for separate spin-up and spin-down ionic pseudopo-

tentials is thus eliminated. What is better, the approach leads to significant

improvement in the transferability of the potentials. It is henceforth under-

standable that this will yield more accurate results both for magnetic and

non-magnetic systems.

Overview of the theory

Louie et. al., in their report[87] have given an example with the normcon-

serving pseudopotential scheme of Hamann, Schliiter, and Chiang[89](HSC).

To start with let me elaborate on the constraints with which the screened

atomic pseudopotentials V lion are constructed. It is to be kept in mind here

that these pseudopotentials are angular-momentum-dependent. The con-

73

straints are -

• The valence eigenvalues from the all-electron calculation and those from

the pseudopotential calculation agree for any configurational prototype.

• The all-electron wave functions and the pseudo-wave functions agree

beyond a chosen core radius, rc.

The two effective properties that are equally desirable and essential come

out from these basic constraints.

• The electrostatic potential produced outside rc, is identical for the all-

electron and the pseudocharge distribution.

• The scattering properties of the all-electron atoms are reproduced with

minimum error as the electronic eigenvalues move away from the pro-

totype atomic levels.

These two properties are claimed to be the reason for ensuring a reasonable

transferabihty of the pseudopotentials. The final bare-ion pseudopotentials,

V lion are extracted from the neutral potentials by subtracting from each neu-

tral V l the Coulomb and exchange and correlation potentials due to the

pseudovalence charge density, ρv(r). As for example, for a given angular

momentum component l and spin component σ, the ionic potential is given

by

V lσion(r) = V lσ(r)− Vee[ρ

v(r)]− Vxc[ρv(r), ξv(r)], (4.2)

where

ξv(r) =ρv+(r)− ρv

−(r)

ρv(r)(4.3)

is the spin polarization of the valence charge with the + and − signs denoting

the spin-up and spin-down electrons respectively. Ionic potentials generated

this way for the nonmagnetic case, i.e. for cases where ξ = O, have been

shown to be highly accurate in many applications[90]. In the above procedure

the basic assumptions are the frozen-core approximation and a decoupling of

the core charge in the determination of the exchange and correlation potential

seen by the valence electrons. At this point let us not forget an important

74

observation about the frozen-core assumption. It, however, implies a single

ionic pseudopotential which is independent of the spin polarization of the

valence states. Hence evidently it is the second approximation that gives rise

to the spin-dependent ionic potentials in previous work. In the HSC approach

the ionic pseudopotentials are dependent on the valence configuration, since

Vxc is a nonlinear function of the charge density and the valence charge does

not cancel in their formalism. It is explicitly written as,

V σxc(ρ

v + ρc, ξ) = [V σxc(ρ

v + ρc, ξ)− V σxc(ρ

v, ξv)] + V σxc(ρ

v, ξv), (4.4)

where

ξ(r) =ρv+(r)− ρv

−(r)

ρv(r) + ρc(r)(4.5)

It is actually clear that in the construction of the ionic potential, the terms

in brackets are included in the ionic potential as part of the core properties

which causes the problem in the transferability of the potential. This feature

is highly undesirable since it reduces the transferability of the potential.

In particular, for magnetic applications, the spin-density distribution of the

electrons can be extremely different both in magnitude and in profile as one

goes from the atomic case to the various condensed matter systems. Hence it

is improbable that any interpolation formula between spin-up and spin-down

potentials generated from atoms will work to a high level of accuracy and

satisfaction. To remove the dependence of the ionic pseudopotential on the

valence charge a simple and straightforward trick can be applied. The form

of the V lσion can be written as -

V lσion(r) = V lσ(r)− Vee[ρ

v(r)]− Vxc[(ρvr + ρcr), ξv(r)]. (4.6)

The total exchange and correlation potential, including the nonlinear core

valence term, is now subtracted out of the neutral potential. A smart and

effective implementation of this theory have been reported to have given

ionic potential highly transferable and essentially independent of the spin

polarization and the prototype atomic configuration. The following figure

with the results illustrate the acceptability of the above mentioned theoretical

75

method. In this regard the basic idea of implementation can be discussed.

Figure 4.1: Atomic term values and total energy differences between para-magnetic and fully spin-polarized atoms. Paramagnetic configurations are3s13p3 for Si and 4d55s1 for Mo. All energies are in eV. ∆E is the totalenergy difference between the paramagnetic and the spin-polarized configu-ration. Superscripts indicate the electron occupation and the + signs denotethe spin configuration for each orbital.

In the employment of the new potential, the core charge should be added

to the valence charge whenever the exchange and correlation potential or

energy is calculated. This core charge remains the same in all applications,

as we are considering things within the frozen core approximation. Hence, in

addition to the usual s, p,and d potentials, we need to retain the core charge

density which is computed once and for all in the same atomic calculations

as the pseudopotentials. It is certain that in an atomic calculation there is

no difficulty in representing the core charge. In a bulk calculation, however,

there are two practical considerations that must be taken care of. In any

pseudopotential calculation there are small, but unavoidable errors in the

calculated valence charge density. Usually this leads to a negligible error in

the total energy, but when the core charge is added, any inaccuracy in the

valence charge density inside the core region is multiplied by the core charge

and the error in the total energy will increase proportionally. To be sure of the

accuracy of the calculation, it is therefore absolutely necessary to treat the

effect of the core charge as a perturbation, and a change in the total energy

76

as a result of this correction should not be large. The above mentioned issues

can be handled by observing that the core charge has significant effect only

where the core and the valence charge densities are of similar magnitude.

One can therefore replace the full core charge density with a partial core

charge density which is equal to the true charge density outside some radius

rc and arbitrary inside. Louie[87] and his colleagues show that rc may be

chosen as the radius where the core charge density is from 1 to 2 times larger

than the valence charge density.

4.3 Tests for the pseudopotentials

The implementation of the PBE pseudopotential on the wavelet basis is

tested on two different levels. With a particular family of PBE pseudopoten-

tial, essentially the Hartwigsen, Goedecker, Hutter (HGH) normconserving

pseudopotentials, the first test were done. This family of pseudopotentials

are known to have the same limitations as that of the HSC pseudopotentials,

i.e. the core electron density is completely neglected for a prototype config-

uration. The results will show the performance of such pseudopotentials. It

can be mentioned here for the sake of information that the results were not

satisfactory and hence an improvement was sought for. The second level of

this test was to account for the limitations that we found to be inherent in

the method from the first test. The NLCC method was found theoretically

suitable for the cause and the next set of tests were done with this new family

of pseudopotentials corrected with the contribution from the core densities.

In the following sections the details of the calculations are explained with

results and analysis. The G2-1 test set has been taken and the atomization

energies for each of the molecules are calculated. The atomization energy

of a molecule E0(M) is defined as the diffrence between the energy of the

molecule ǫ0(M) and the sum of the energies of the constituent atoms ǫ0(X),

E0(M) =∑

species

nǫ0(X)− ǫ0(M). (4.7)

77

For all the atoms spin polarized calculations were done. It was taken strictly

into consideration that atoms were calculated in a symmetry broken ground

state. This state was found to be lower in energy than the spherically high-

symmetry solution. In the code care was taken as to satisfy the Hund’s rule

of electronic configuration for each constituent atom of each molecule. In the

following section let us take a look at a brief overview of Hund’s rule.

4.3.1 Hund’s rule

The rules formulated by Hund help to determine the total spin S, the total

spatial angular momentum L, and the total angular momentum J = L + S

of the ground state of an atom.

• If for a single atom there can be many electron states, the ground state

will be determined by the one having the largest total spin S.

• Among the states with the largest total spin, the ground state is the

one with the highest angular momentum quantum number L.

• For a shell less than being half-filled that ground state is the one with

the lowest value of J = L + S.

• For a shell more than being half-filled that ground state is the one with

the highest value of J = L + S.

These are the rules which are applied to the atoms for the calculations un-

dertaken in this work. But to apply these rules to the molecules care must

be taken so as to get the correct symmetry broken ground state. Hunds first

rules are reported to be good enough for most molecules. Most molecules of

interest have closed shell ground states, but there are a few exceptions. In

those cases Hund’s rule predicts that triplet lies below the singlet. In all the

calculations done in this work the Hund’s rules are applied with rigor and

care.

78

4.3.2 Calculational parameters

The main motivation for these calculations is to use the wavelet basis set and

to see the validity of the corresponding implementation of the NLCC within

the PBE pseudopotentials. In order to be highly accurate with atomization

energies for these calculations the parameters viz. hgrid and crmult for each

atom are tested for different values. Genovese et. al.[43] have mentioned the

theoretical convergence of the physical quantities calculated with respect to

these parameters. hgrid is the grid spacing and crmult is a parameter which

determines the extent of resolution zone that contains the exponentially de-

caying tails of the wave functions. Care has been taken to find the correct

combination of hgrid and crmult for each atom in the list of the G2-1 test

set of molecules. The hgrid and crmult are chosen in such a fashion that the

error in the total energy incurred in the calculation must not exceed 10−5

Hartrees, which is close to an error of .25 meV. Now from figure 4.2 it is

evident that for the atom hydrogen hgrid value of 0.30 and a crmult value

of 6 would give us results which are satisfactorily within our accuracy need.

Table 1 lists the values of the two parameters used for the elements in the

set.

Once the basic parameters to work with are acheived the main calculation

for all the atoms for the atomization energies are launched. As mentioned

earlier all the energies of the atoms are calculated in their spin polarized

state.

4.3.3 Atomization energy of molecules using PBE

The first test is performed with the PBE functionals in comparison with

the results of the plane wave methods as reported in the works of Paier[85]

et. al. From the figure 4.3 there are few interesting things that can be

understood. In a general statement one may consider the PBE functionals as

implemented on the wavelet basis set tends to underestimate the atomization

energy values as calculated from the PAW and G03 codes. Of course there

are atoms which have given excellent results in comparison to the all electron

calculations, but this is not the kind of test that will testify for the accuracy

79

hgrid in Bohr

crmult

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

err

or

in e

ne

rgy in

Ha

rtre

e

for Hydrogen atom

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

45

67

89

10

err

or

in e

ne

rgy in

Ha

rtre

e

0

0.0005

0.001

0.0015

0.002

0.0025

Figure 4.2: The error in total energy calculation of an H atom as a functionof the hgrid and crmult. The drawn dotted contour corresponds to the ∆Evalue of 10−7 Hartree.

of the impletation of PBE on the wavelet basis set. The main motive of this

work is to validate the implementation of the NLCC and its transferability.

The reasons for some atoms to be giving particularly good results can be

attributed to mainly two facts. One may believe as a primary reason that

the approximation to the densities, without taking into account the core

charges, were good enough for those prototypes. Again for molecules where

there are two or more atoms, respective errors in the calculation of the total

energies of each constituent atom cancelled the overall error. But obviously

in general the PBE functional as implemented on the wavelet basis set tends

to underestimate the atomization energy value. A careful observation of

the graph would help us to understand better the sources of errors. The

80

Table 4.1: The elements and the corresponding hgrids and crmults.

Elements hgrid crmultH 0.350 6Li 0.300 6Be 0.250 6C 0.250 6N 0.250 6O 0.200 6F 0.150 7Na 0.200 8Si 0.350 6P 0.350 7S 0.300 7Cl 0.250 7

molecules like O2, CO2, N2H4, N2, H2O2 have suffered the highest deviations

from the all electron calculations. The presence of oxygen in many of those

molecules hints towards the electronegativity of the atoms as well. Where as

the role of hydrogen may be excluded as it has only one electron and hence

there arises no question of the core charge. The nitrogen atom on the other

hand poses some significant problems. The reason to this may be because

of the strength of bonding of the nitrogen valence electrons. To associate

one nitrogen atom to any other atoms to form a molecure the core charge

density of the nitrogen might be perturbed and an inadequate representation

of that in the pseudopotential might result in the type of errors that are

exhibited in the figure. The following table shows the values as calculated

with BigDFT in comparison with reported values of VASP and G03 along

with the experimental values.

The average deviation of our results with this PBE pseudopotentials from

that of the G03 calculations is 5.197 kcal/mol and from that of Vasp is 5.015

kcal/mol. It is expected to improve upon the implementation of the non

linear core correction as with that formalism we hope to get closer to the

results obtained by all electron methods.

81

Table 4.2: Electronic atomization energies for the molecules in the G2-1 testset using the PBE functional. Energies are in kcal/mol.

Moleulcues AEPBE AEexpt ∆Expt− BigDFT% AEV asp AEG03 ∆V asp−BigDFT ∆G03−BigDFT

LiH 53.491 58.000 7.774 53.500 53.500 0.009 0.009BeH 55.660 48.000 15.958 55.500 55.600 -0.160 -0.060CH 82.819 84.000 1.406 84.700 84.800 1.881 1.981OH 105.566 107.000 1.340 109.700 110.100 4.134 4.534O2 130.381 118.000 10.492 143.300 144.000 12.919 13.619F2 44.641 38.000 17.476 52.600 53.000 7.959 8.359Li2 19.935 26.000 23.327 19.900 20.100 -0.035 0.165CO 254.750 261.000 2.395 268.600 269.100 13.850 14.350LiF 134.071 139.000 3.546 138.400 139.000 4.329 4.929SiO 190.156 191.000 0.442 195.600 196.600 5.444 6.444HF 136.815 142.000 3.651 141.500 142.200 4.685 5.385CN 185.236 179.000 3.484 197.500 197.700 12.264 12.464N2 226.867 227.000 0.059 243.700 243.900 16.833 17.033Cl2 64.757 57.000 13.609 65.800 65.800 1.043 1.043ClO 75.938 62.000 22.481 81.600 81.500 5.662 5.562HCl 105.664 107.000 1.249 106.300 106.500 0.636 0.836ClF 67.571 62.000 8.985 72.300 72.500 4.729 4.929Na2 17.497 19.000 7.911 17.700 18.100 0.203 0.603NO 157.290 153.000 2.804 172.000 172.500 14.710 15.210Si2 78.775 74.000 6.453 81.300 81.400 2.525 2.625NH 86.005 82.000 4.884 88.600 88.600 2.595 2.595P2 121.487 116.000 4.730 121.500 121.700 0.013 0.213S2 114.660 98.000 17.000 115.400 115.200 0.740 0.540SO 135.196 122.000 10.816 141.500 141.300 6.304 6.104CS 173.745 172.000 1.015 179.500 179.500 5.755 5.755CH3 306.418 306.000 0.137 309.700 310.100 3.282 3.682NH2 182.999 182.000 0.549 188.700 188.900 5.701 5.901H2O 225.689 233.000 3.138 233.700 234.500 8.011 8.811CO2 392.546 392.000 0.139 415.400 416.500 22.854 23.954SO2 269.089 253.000 6.359 281.100 280.700 12.011 11.611SH2 180.978 182.000 0.562 182.000 182.200 1.022 1.222H2O2 264.498 268.000 1.307 281.600 282.600 17.102 18.102HCN 311.872 313.000 0.360 326.300 326.500 14.428 14.628HCO 280.544 279.000 0.553 294.900 295.500 14.356 14.956HOCl 167.033 165.000 1.232 175.200 175.700 8.167 8.667NH3 293.184 297.000 1.285 301.700 302.300 8.516 9.116PH2 155.144 153.000 1.401 154.500 154.600 -0.644 -0.544PH3 239.176 241.000 0.757 239.000 239.300 -0.176 0.124NaCl 93.077 99.000 5.983 93.600 94.500 0.523 1.423CH4 413.802 420.000 1.476 419.600 420.200 5.798 6.398

SiH2(1A1) 147.577 154.000 4.171 147.900 148.000 0.323 0.423SiH3 222.992 226.000 1.331 222.200 222.600 -0.792 -0.392SiH4 313.536 324.000 3.230 313.300 313.700 -0.236 0.164C2H2 403.066 404.000 0.231 414.500 415.100 11.434 12.034C2H4 559.583 562.000 0.430 571.000 571.900 11.417 12.317C2H6 704.483 711.000 0.917 716.000 717.100 11.517 12.617H2CO 371.528 376.000 1.189 385.000 386.300 13.472 14.772N2H4 431.551 437.000 1.247 452.700 453.700 21.149 22.149

CH2(1A1) 174.512 182.000 4.114 178.800 179.100 4.288 4.588CH2(3B1) 193.571 189.000 2.419 194.400 194.600 0.829 1.029CH3OH 505.051 513.000 1.550 519.300 520.400 14.249 15.349CH3Cl 392.693 395.000 0.584 399.400 400.200 6.707 7.507CH3SH 470.621 473.000 0.503 477.800 478.600 7.179 7.979Si2H6 519.258 533.000 2.578 519.500 520.400 0.242 1.142

SiH2(3B1) 125.044 131.000 4.547 131.300 131.800 6.256 6.756

82

-20

-10

0

10

20

LiH

BeH CH

OH

O2

F2

Li2

CO

LiF

SiO HF

CN N2

Cl2

ClO

HC

lC

lFN

a2 NO

Si2

NH P2

S2

SO

CS

CH

3N

H2

H2O

CO

2S

O2

SH

2H

2O2

HC

NH

CO

HO

Cl

NH

3P

H2

PH

3N

aCl

CH

4S

iH2(

1A1)

SiH

3S

iH4

C2H

2C

2H4

C2H

6H

2CO

N2H

4C

H2(

1A1)

CH

2(3B

1)C

H3O

HC

H3C

lC

H3S

HS

i2H

6S

iH2(

3B1)

Diff

eren

ce in

the

atom

izat

ion

ener

gy fr

om th

e G

03 c

alcu

latio

ns

molecules of G21 set

pbewithoutnlcc-wavelet-basis-setpawG03

Figure 4.3: Electronic atomization energies for the molecules in the G2-1test set using the PBE functional. Energies are plotted in kcal/mol. The allelectron values from the Gaussian calculations are taken as the standard.

Geometries of the molecules

To complete the study with the PBE functionals the geometry of the molecules

were also investigated. A set of sixteen di-atomic molecules have been taken.

The molecules are made to go geometrical relaxation and then the bond

lengths are noted down. The following plot shows the variation of the bond

lengths with respect to the all electron calculations. The break criterion for

the geometry optimization in BigDFT was set to 10−5Hartree/Bohr which

is equivalent to 0.0005eV/A.

The table 4.3 and the figure 4.4 are an illustration of the fact that how

well the bondlength results have matched with the PAW and Gaussian cal-

culations. The bond lengths are already very good without the NLCC cal-

culations. One can although see a general trend. The bond length values

from all the three different theoretical calculations tend to overestimate the

83

Table 4.3: Tabular representation of theoretical and experimental bondlengths of 16 diatomics chosen from the reduced G2 data set. Bondlengthsare in angstroms.

Molecules Expt. PAW G03 BigDFT abs(∆BigDFT −G03)

in A in A in A in A in A×103

BeH 1.343 1.354 1.353 1.354 1CH 1.120 1.136 1.136 1.136 0Cl2 1.988 1.999 2.004 1.994 10ClF 1.628 1.648 1.650 1.645 5ClO 1.570 1.576 1.577 1.575 2CN 1.172 1.173 1.174 1.173 1CO 1.128 1.136 1.135 1.135 0F2 1.412 1.414 1.413 1.416 3FH 0.917 0.932 0.930 0.932 2HCl 1.275 1.287 1.288 1.286 2Li2 2.673 2.728 2.728 2.728 0LiF 1.564 1.583 1.575 1.575 0LiH 1.595 1.604 1.604 1.604 0N2 1.098 1.103 1.102 1.102 0O2 1.208 1.218 1.218 1.220 2Na2 3.079 3.087 3.076 3.070 6

84

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

BeH CH Cl2 ClF ClO CN CO F2 FH HClLi2 LiF LiH N2 O2 Na2

Diff

eren

ce o

f the

nlc

c b

ond-

legt

h va

lues

with

PA

W G

03 a

nd e

xpt.

List of selected diatomic molecules

pawg03pbe

expt refernce

Figure 4.4: Graphical representation of theoretical and experimental bondlengths of 16 diatomics chosen from the reduced G2 data set. Bondlengthdifferences are in angstroems.

experimental bond lengths. For most of the elements we find that the bond

lengths are very good accordance with the G03 calculations. As we can see

the problems for Paw with some of the molecules like LiF do not arise in

our case. The reason for the agreement for the bond lengths even with in-

accurate atomization energies can be explained as the following. Clearly for

the energies we see a problem with the reference energies of single atoms.

They are shifted in general from the accurate values. But since the bond

length calculations depend on the force calculations, which is essentially the

derivative of the energy surface. It appears that although the energy surface

is shifted by a value yet the shape of the surface is calculated with relatively

more accuracy and hence the force calculations are more accurate. Care

was taken by Paier[85] et. al. to minimise the dicrepancies. For example

in particular, the box size was increased, dipole correction was applied, and

85

the energy cutoff was increased. Yet the reason for this inaccuracy remains

open. It is also to be kept in mind here that for Li all electrons were treated

as valence electrons. There is another problem to be looked after both for

the PAW calculations as well as the wavelet calculations. The problem is

with the Na2 molecule. The Na2 molecule demand a tight convergence cri-

terion for the geometry relaxation because of the weak interaction between

the constituent Na atoms.

4.3.4 Atomization energy of molecules using PBE with

NLCC

Having seen the performance of PBE as implemented on the wavelet basis

(table 4.4 and figure 4.5) we are tempted to look for betterment in terms of

accuarcy towards the all electron calculation. With that view as discussed

in the precious sections the non linear core correction is the one interesting

way that could be thought of. A new family of pseudopotential is available

now under the PBE formalism with NLCC implemented in it. There are

parameters that can be played around with for respective elements to get

the most accurate and desirable results. Of course for atoms like hydrogen

and lithium these corrections are meaningless. Yet as it will be evident from

the following table and figures that considerable amount of improvement is

obtained with this implementation.

A single glance would show that the implementation of the NLCC in the

pseudopotentials have inproved the results by a large margin. Here also the

convergence parameters are kept the same as before for each prototype con-

figuration.

The significant improvement in the results can also be seen in the values of

mean absolute deviation of the atomization energies of BigDFT-Vasp and

BigDFT-G03. The average deviation of our results with this PBE pseu-

dopotentials from that of the G03 calculations is 1.03 kcal/mol and from

that of Vasp is 1.05 kcal/mol. Hence we can say at this point that in terms

of closeness to the calculations of the PAW and G03 the NLCC improves

the previous calculations by a factor of 5. Indeed this improvement is very

86

Table 4.4: Electronic atomization energies for the molecules in the G2-1test set using the PBE functional with NLCC correction. Energies are inkcal/mol.

Moleulcues AEPBE AENLCC AEexpt ∆Expt− BigDFT% AEV asp AEG03 ∆V asp−BigDFT ∆G03−BigDFT

LiH 53.491 53.489 58.000 7.778 53.500 53.500 0.011 0.011BeH 55.660 55.657 48.000 15.952 55.500 55.600 -0.157 -0.057CH 82.819 84.257 84.000 0.306 84.700 84.800 0.443 0.543OH 105.566 109.842 107.000 2.656 109.700 110.100 -0.142 0.258O2 130.381 144.184 118.000 22.190 143.300 144.000 -0.884 -0.184F2 44.641 54.495 38.000 43.409 52.600 53.000 -1.895 -1.495Li2 19.935 19.935 26.000 23.328 19.900 20.100 -0.035 0.165CO 254.750 270.133 261.000 3.499 268.600 269.100 -1.533 -1.033LiF 134.071 140.275 139.000 0.917 138.400 139.000 -1.875 -1.275SiO 190.156 200.394 191.000 4.918 195.600 196.600 -4.794 -3.794HF 136.815 142.809 142.000 0.570 141.500 142.200 -1.309 -0.609CN 185.236 197.722 179.000 10.459 197.500 197.700 -0.222 -0.022N2 226.867 243.491 227.000 7.265 243.700 243.900 0.209 0.409Cl2 64.757 66.572 57.000 16.793 65.800 65.800 -0.772 -0.772ClO 75.938 82.327 62.000 32.786 81.600 81.500 -0.727 -0.827HCl 105.664 107.164 107.000 0.153 106.300 106.500 -0.864 -0.664ClF 67.571 73.494 62.000 18.539 72.300 72.500 -1.194 -0.994Na2 17.497 17.481 19.000 7.995 17.700 18.100 0.219 0.619NO 157.290 171.982 153.000 12.407 172.000 172.500 0.018 0.518Si2 78.775 79.815 74.000 7.858 81.300 81.400 1.485 1.585NH 86.005 87.747 82.000 7.008 88.600 88.600 0.853 0.853P2 121.487 124.118 116.000 6.999 121.500 121.700 -2.618 -2.418S2 114.660 114.671 98.000 17.011 115.400 115.200 0.729 0.529SO 135.196 140.503 122.000 15.166 141.500 141.300 0.997 0.797CS 173.745 179.232 172.000 4.205 179.500 179.500 0.268 0.268CH3 306.418 309.409 306.000 1.114 309.700 310.100 0.291 0.691NH2 182.999 188.102 182.000 3.353 188.700 188.900 0.598 0.798H2O 225.689 235.435 233.000 1.045 233.700 234.500 -1.735 -0.935CO2 392.546 416.205 392.000 6.175 415.400 416.500 -0.805 0.295SO2 269.089 275.731 253.000 8.985 281.100 280.700 5.369 4.969SH2 180.978 181.665 182.000 0.184 182.000 182.200 0.335 0.535H2O2 264.498 283.128 268.000 5.645 281.600 282.600 -1.528 -0.528HCN 311.872 326.192 313.000 4.215 326.300 326.500 0.108 0.308HCO 280.544 293.879 279.000 5.333 294.900 295.500 1.021 1.621HOCl 167.033 176.017 165.000 6.677 175.200 175.700 -0.817 -0.317NH3 293.184 301.434 297.000 1.493 301.700 302.300 0.266 0.866PH2 155.144 155.808 153.000 1.836 154.500 154.600 -1.308 -1.208PH3 239.176 241.186 241.000 0.077 239.000 239.300 -2.186 -1.886NaCl 93.077 95.034 99.000 4.006 93.600 94.500 -1.434 -0.534CH4 413.802 419.374 420.000 0.149 419.600 420.200 0.226 0.826

SiH2(1A1) 147.577 148.582 154.000 3.518 147.900 148.000 -0.682 -0.582SiH3 222.992 223.630 226.000 1.048 222.200 222.600 -1.430 -1.030SiH4 313.536 315.582 324.000 2.598 313.300 313.700 -2.282 -1.882C2H2 403.066 414.388 404.000 2.571 414.500 415.100 0.112 0.712C2H4 559.583 570.842 562.000 1.573 571.000 571.900 0.158 1.058C2H6 704.483 715.702 711.000 0.661 716.000 717.100 0.298 1.398H2CO 371.528 387.062 376.000 2.942 385.000 386.300 -2.062 -0.762N2H4 431.551 448.464 437.000 2.623 452.700 453.700 4.236 5.236

CH2(1A1) 174.512 178.250 182.000 2.060 178.800 179.100 0.550 0.850CH2(3B1) 193.571 194.145 189.000 2.722 194.400 194.600 0.255 0.455CH3OH 505.051 519.241 513.000 1.217 519.300 520.400 0.059 1.159CH3Cl 392.693 399.551 395.000 1.152 399.400 400.200 -0.151 0.649CH3SH 470.621 476.638 473.000 0.769 477.800 478.600 1.162 1.962Si2H6 519.258 523.246 533.000 1.830 519.500 520.400 -3.746 -2.846

SiH2(3B1) 125.044 124.388 131.000 5.047 131.300 131.800 6.912 7.412

87

-20

-10

0

10

20

LiH

BeH CH

OH

O2

F2

Li2

CO

LiF

SiO HF

CN N2

Cl2

ClO

HC

lC

lFN

a2 NO

Si2

NH P2

S2

SO

CS

CH

3N

H2

H2O

CO

2S

O2

SH

2H

2O2

HC

NH

CO

HO

Cl

NH

3P

H2

PH

3N

aCl

CH

4S

iH2(

1A1)

SiH

3S

iH4

C2H

2C

2H4

C2H

6H

2CO

N2H

4C

H2(

1A1)

CH

2(3B

1)C

H3O

HC

H3C

lC

H3S

HS

i2H

6S

iH2(

3B1)

Diff

eren

ce in

the

atom

izat

ion

ener

gy fr

om th

e G

03 c

alcu

latio

ns

molecules of G21 set

without-nlccnlccpawG03

Figure 4.5: Electronic atomization energies for the molecules in the G2-1 test set using the PBE functional with NLCC. Energies are plotted inkcal/mol. The all electron values from the Gaussian calculations are takenas the standard.

significant.

Geometries of the molecules with NLCC

To complete the study with the PBE functionals with NLCC the geometry

of the molecules were also investigated in the same spirit as before. the same

set of sixteen diatomic molecules have been taken. The following plot in

figure 4.6 shows the improvement in the variation of the bond lengths with

respect to the all electron calculations. The break criterion for the geometry

optimization in BigDFT was set to 10−5Hartree/Bohr which is equivalent

to 0.0005eV/A which is same as the calculations without the NLCC.

88

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

BeH CH Cl2 ClF ClO CN CO F2 FH HClLi2 LiF LiH N2 O2 Na2

Diff

eren

ce o

f the

nlc

c b

ond-

legt

h va

lues

with

PA

W G

03 a

nd e

xpt.

List of selected diatomic molecules

pawg03

without-nlccnlcc

Figure 4.6: Graphical representation of theoretical and experimental bondlengths of 16 diatomics chosen from the reduced G2 data set with NLCC.Bondlength differences are in angstroms.

The bond lengths are observed to be more accurate as the figure and ta-

ble above suggest. It is also noteworthy here that in comparison to PAW

calculations our results have been found closer to that of the all electron

gaussian calculations for most of the molecules. This indicates the extent of

the success of the implementation of the NLCC in the pseudopotentials.

A closer analysis

Although as mentioned earlier the results have improved considerably with

the implementation of the NLCC yet a closer look into the graphical repre-

sentation (figure 4.7) would reveal some remnant discrepancies. The reason

89

Table 4.5: Tabular representation of theoretical and experimental bondlengths of 16 diatomics chosen from the reduced G2 data set with NLCC.Bondlengths are in angstroms.

Molecules Expt. PAW G03 BigDFT(NLCC)BeH 1.343 1.354 1.353 1.354CH 1.120 1.136 1.136 1.137Cl2 1.988 1.999 2.004 2.003ClF 1.628 1.648 1.650 1.649ClO 1.570 1.576 1.577 1.578CN 1.172 1.173 1.174 1.173CO 1.128 1.136 1.135 1.135F2 1.412 1.414 1.413 1.415FH 0.917 0.932 0.930 0.931HCl 1.275 1.287 1.288 1.287Li2 2.673 2.728 2.728 2.728LiF 1.564 1.583 1.575 1.575LiH 1.595 1.604 1.604 1.604N2 1.098 1.103 1.102 1.102O2 1.208 1.218 1.218 1.220Na2 3.079 3.087 3.076 3.070

90

-6

-4

-2

0

2

4

6

LiH

BeH CH

OH

O2

F2

Li2

CO

LiF

SiO HF

CN N2

Cl2

ClO

HC

lC

lFN

a2 NO

Si2

NH P2

S2

SO

CS

CH

3N

H2

H2O

CO

2S

O2

SH

2H

2O2

HC

NH

CO

HO

Cl

NH

3P

H2

PH

3N

aCl

CH

4S

iH2(

1A1)

SiH

3S

iH4

C2H

2C

2H4

C2H

6H

2CO

N2H

4C

H2(

1A1)

CH

2(3B

1)C

H3O

HC

H3C

lC

H3S

HS

i2H

6S

iH2(

3B1)

Diff

eren

ce in

the

atom

izat

ion

ener

gy fr

om th

e G

03 c

alcu

latio

ns

molecules of G21 set

nlccpawG03

Figure 4.7: Electronic atomization energies for the molecules in the G2-1 testset using the PBE functional with only NLCC correction . Energies are inkcal/mol. This is the zoomed version of figure 4.5.

for these discrepancies are to be sought after in detail.

In figure 4.8 what we get to see is the respective contribution of each el-

ement in a particular zone of absolute deviation from the G03 values. In

the figure Di-j is a zone of deviation in kcal/mol. Hence D0-1, D1-2, D2-4,

D4-7 are the deviation ranges of 0-1 kcal/mol, 1-2 kcal/mol, 2-4 kcal/mol

and 4-7 kcal/mol respectively. We can exclude hydrogen from our analysis

because of two main reasons. First of all, it has only one electron and hence

there is no question of fixing a special core density. Secondly it is the most

frequent element in the entire list. So it is quite evident that its contribution

is show in almost every range. The ranges which worry us the most are D2-4

and D4-7. Although we see some contribution of oxygen in those ranges,

91

0

5

10

15

20

25

30

35

40

45

Be C Cl F H Li N Na O P S Si Si

num

ber

of c

ontr

ibut

ions

to th

e co

rres

pond

ing

erro

r ra

nge

constituent elements

D0-1D1-2D2-4D4-7

Figure 4.8: Contribution of each element in a particular zone of absolutedeviation from the G03 values for the NLCC calculations.

a quick look at the table above would qualify oxygen as another of a fre-

quent element. Its association with elements like silicon and sulphur would

have forced its presence in those ranges of deviation. To this we come to

the elements like silicon, sulphur, phosphorous and nitrogen. The problem

with nitrogen is probably the same as discussed above. Its association with

hydrogen in N2H4 might have perturbed the core density which is still ill

represented in NLCC formalism. On the other hand the cases for silicon and

sulphur are not that straightforward. Silicon is semimetallic. An intuitive

approach would call for a smaller rc. The radius rc as described in the theory

section is an important parameter in this pseudopotential approach. Another

solution could be to play around with the amount of core-density to be used

in the NLCC part. At the end a general algorithm is what we have tried to

92

find that will be applied for all the elements in the same spirit to give more

accurate results. By this I mean we need to get rid of the deviation ranges

D2-4 and D4-7. The fact the silicon is indeed a bit tricky to handle, let

0

10

20

30

40

50

60

Be C Cl F H Li N Na O P S Si Si

num

ber

of c

ontr

ibut

ions

to th

e co

rres

pond

ing

erro

r ra

nge

constituent elements

D0-1D1-2

Figure 4.9: Contribution of each element in a particular zone of absolutedeviation from the G03 values for the PAW calculations.

us look in figure 4.9 where I have taken the data from the work of Paier[85]

and his collaborators. Here also we can see that silicon is found to be among

the very few to have contributed to the relatively higher deviation zone. In

this figure though we see contributions of carbon which in turn is not signifi-

cant with the NLCC calculations. Hence there is an indication regarding the

beahaviour of silicon in this particular picture. On the other hand in the

calculations with NLCC we have to look for other discrepancies arising from

sulphur and phosphorous. The problem with the phosphorous is indeed that

of a smaller rc. Calculations with smaller rc have yielded improved results

93

but the physics behind this improvement is still not better understood. The

question which is open at the moment is how to tackle the case of sulphur in

this general approach to create the pseudopotentials with NLCC.

4.3.5 Similar effort with Linear Spin Density Approx-

imation(LSDA)

Since the results with the calculations with the NLCC corrections to the

PBE pseudopotentials were found encouraging, efforts were made to do the

same under the LSDA formalism. The expected performance of those LSDA

pseudopotentials were on the same lines as we experienced with the PBE

+ NLCC pseudopotentials. In the article by Fournier[91] et. al. the cal-

-40

-30

-20

-10

0

10

20

30

CH

OH

O2

F2

Li2

CO

LiF

SiO CN N2

Cl2

ClO ClF

NO

Si2

NH P2

S2

SO

CS

CH

3N

H2

CO

2S

O2

H2O

2H

CN

HC

OH

OC

lP

H2

NaC

lC

H4

SiH

2(1A

1)S

iH2(

3B1)

SiH

3S

iH4

C2H

2C

2H4

H2C

ON

2H4

CH

2(1A

1)C

H2(

3B1)

CH

3OH

CH

3Cl

CH

3SH

Diff

eren

ce in

the

atom

izat

ion

ener

gy fr

om th

e LS

DA

cal

cula

tions

molecules of G1 set

abs-error lsda-BigDFt without nlccabs-error lsda-BigDFt with nlcc

Figure 4.10: Deviation of the BigDFT values from

culations of the atomization energies are being reported. They have calcu-

94

lated the atomization energies by different theories like Kohn–Sham density

functional theory (KS-DFT) with local spin density approximation (LSDA),

two KS-DFT gradient-corrected methods, one hybrid Hartree–Fock/KS-DFT

method similar to B3LYP, and the ab initio extrapolation procedures G1 and

G2. They have reported that the empirical corrections improved the LSDA

results greatly, while the other theories were improved slightly or not at all.

We have tried to compare our results with LSDA and NLCC with the results

of this report. In figure 4.10 the deviation of our calculations with their re-

ported values are plotted for both the cases. The two cases are viz. LSDA

without NLCC and LSDA with NLCC. as evident from the graph the results

are not encouraging. But the reason for such discrepancy can be thought

of as the error coming from the zero point energy which is neglected in the

cases of LSDA calculations.

4.4 Conclusion

In this part of the research we have tried to find an effective way to im-

plement the NLCC in the PBE and LSDA pseudopotentials on a wavelet

basis. The idea was to calculate the atomization energies to a ceratin de-

gree of accuracy and hence to use the pseudopotentials in further research.

Since wavelet basis sets have shown great promist in the DFT calculations the

need to implement various pseudopotentials on the wavelet basis set was felt.

As of now as we have shown that the results with the NLCC implemented

PBE pseudopotentials show good promise. Of course there is still room for

improvement. And also the need to understand the deeper physics for few

elements in the pseudopotential approach is felt. It must be highlighted that

the non-conserving pseudopotentials can be as precise as PAW if NLCC is

taken into acoount. We have also shown that the NLCC pseudopotential

seemingly improve the overall calculations by a factor of 5, which is indeed

a promising step in way of further research. Another important fact that

is revealed in the above analysis is the independence of the accuracy of the

geometrical structures from the jargon of NLCC in the pseudopotentials. By

this we mean that even with pseudopotentials where the entire core density

95

is considered to be constant the geometries of the molecules are correctly

predicted and understood. Having said that, we must also notice that im-

plementation of NLCC improved the geometry part as well. The need to

understand the LSDA formalism within the framework of NLCC is felt very

deeply. At present the question is open as to why the discrepancy of the

results would arise from the calculations of LSDA+NLCC as implemented

on the wavelet basis.

In the next chapter we will go into the main part of the thesis, the treatment

of the charged defects in silicon nanoclusters. Although those calculations

are done without NLCC as the work was simultaneous yet for future cal-

culations of charged defects one is tempted to use these NLCC corrected

pseudopotentials. In fact the next calculations without NLCC are sure to

give the geometries to a pretty high accuracy which is another reason to use

such NLCC uncorrected pseudopotentials.

96

Chapitre 5

Ce chapitre aborde le sujet des defauts charges dans le silicium. Jusqu’a

present, les defauts charges ont principalement ete etudies en conditions

periodiques (PBC). En effet, l’approche PBC est parfaite pour simuler un

solide infini. Mais, comme je le montre dans ce chapitre, elle apporte aussi

des artefacts. En particulier dans le traitement de l’electrostatique, ou des

traitements mathematiques important doivent etre utilises pour supprimer

les interactions non desirees entre repliques. Notre approche vise a simuler

correctement un defaut charge, tout en conservant de bonne proprietes pour

le materiau massif. Elle consiste a simuler le defaut dans un nano-agregat.

Le traitement de l’electrostatique est correct dans un nano-agregat et les

resultats obtenus peuvent etre extrapoles au materiau massif, comme il est

montre dans ce chapitre. Les perspectives de cette methode sont aussi

abordees ici.

97

Chapter 5

Charged Defects in Silicon

Nanoclusters

5.1 Introduction

The defects as it is mentioned in the introductory chapters are studied ex-

tensively under the formalism of DFT. It may be repeated as an emphasis

here as well that these calculations complement experiment in all possible

ways. They not only help to interpret experimental findings and link them

to an atomistic model of the relevant defects, but also provide additional

data such as formation energies, geometric structures, or the character of

the wavefunctions that cannot be obtained with state-of-the-art experimen-

tal methods. It is also worth mentioning again that the defect is usually

modeled in a supercell, consisting of the defect surrounded by many other

atoms of the host material, which is then repeated periodically throughout

space. The treatment is formally known as the Periodic Boundary Condi-

tion(PBC) approach. There have been reports[92] of the effectiveness of this

approach. However, it must be kept in mind that the use of supercells im-

plies that the isolated defect is replaced by a periodic array of defects. Such

a periodic array contains large imagineary defect concentrations, resulting in

artificial interactions between the defects that cannot be neglected by any

means. These interactions include overlap of the wavefunctions, elastic in-

98

teractions, and (in the case of charged defects) electrostatic interaction. A

large variety of approaches to control electrostatic artifacts exists in the lit-

erature, as has been reviewed recently by Nieminen[93]. As I have mentioned

in the earlier chapter (state-of-the-art chapter) supercell calculations for the

charged systems must always include a compensating background charge,

since the electrostatic energy of a system with a net charge in the unit cell

diverges[62][66]. The most common practice is to include a homogeneous

background, which is equivalent to setting the average electrostatic potential

to zero. By increasing the supercell lattice constant L, the isolated defect

limit can be realized in principle for L → ∞. Again it is to be remembered

here that the defect energy converges pretty slowly with respect to L (to be

precise it is L−1). The origin of this effect lies in the electrostatic interaction

of the defect with its periodic images which is unphysical. There is of course

an interaction of the charged defects with the compensating background. Its

magnitude can be estimated from the Madelung energy of an array of point-

charges with neutralizing background[62]. Makov and Payne[66] proved for

isolated ions that the quadrupole moment of the charge distribution gives rise

to a further term scaling like L−3. For realistic defects in condensed systems,

however, such corrections, scaled by the macroscopic dielectric constant ǫ to

account for screening, do not always improve the convergence[94][73]. As

mentioned in the chapter of the state-of-the-art there are other methods to

deal with the electrostatics of these charged defects amidst the image charges

and the background compensation in the PBC framework. All the calcula-

tion reported here are done with the DFT code BigDFT with the parameters

converged for silicon (discussed in the previous chapter). And all the calcu-

lations are spin average. The energy convergence criterion was kept to 10−4

Hartree and the force convergence criterion was kept to 10−5 Hartree/Bohr.

The total energy convergence criterion is based on the norm of the gradient

of the wave-function in the convergence cycle. In the following figure 5.1 and

figure 5.2 it shown in detail how the periodic extension of the supercell brings

in the unphysical elastic and electrostatic interactions between the real defect

and the imagineary defects. In this thesis a new approach is undertaken with

a view to study the charged defects under a simple yet correct electrostatic

99

Figure 5.1: A schematic representation of the image charges around the realdefect in PBC. The red dots inside the highlighted box are the real defects(charged or uncharged) and the rest are the product of the periodic extensionalong all directions.

model. The other method can be broadly classified as the cluster method.

In the cluster approach, a reasonably large cluster of atoms truncated from

the bulk with a defect at the center is used to simulate the bulk defect. For

this work we have created an isolated cluster of silicon atoms saturated by

hydrogen at the surface. Recent advances in electronic structure algorithms,

such as real-space methods[95][96] bring a new perspective to theoretical in-

vestigations of defects in materials in this cluster approach. Over the last few

years, these methods with the use of ab initio pseudopotentials have proven

their efficiency and accuracy for microscopic understanding of many physical

systems. In the works of Serdar Ogut and his collaborators reports have been

made on atomic and electronic structures of neutral and positively charged

monovacancies in bulk Si, investigated from first principles using a cluster

method[97]. In their work they have done calculations in real space on bulk-

terminated clusters containing up to 13 shells around the vacancy (about

200 Si atoms). Vacancy-induced atomic relaxations, Jahn-Teller distortions,

vacancy wave-function characters, and relaxation and reorientation energies

are calculated as a function of the cluster size and compared with available

100

Figure 5.2: A schematic representation of the elastic and electrostatic inter-actions between the real defect and the image defects PBC.

experimental data. But no reports have been made since then about the

applicability of this method in calculating the formation energy of vacancies

along with the formulation of a realistic charged-defect electrostatics. There

are two things which are very important in this approach. The first one is the

surface to bulk interaction as shown in figure 5.4. Since it is an attempt to

simulate bulk with a finite system we will definitely have the finite size effect

which can tracked down by the proper electrostatic analysis. This surface

electron density would essentially interact with the perturbed electron den-

sity around the vacancy to create a new phenomenon of trapped charges, as

will be shown in the following sections. This phenomenon of trapped charge

is the second aspect of this methodology. Again in our analytical model we

101

Figure 5.3: A schematic representation of the cluster method. The clustersare pacified by hydrogen atoms at the surface. The presence of only onearray of red dots inside the box ensures the simplicity and realness of thedefect representation.

would assume the clusters to be spherical, which in reality is true to a large

extent for really large clusters. But as shown in the schematic figure the clus-

ters are, in reality, polygons of some odd shape. Hence there is an element

of shape factor in this analysis as well.

5.2 Comparison of formulae (PBC vs. FBC)

5.2.1 PBC formalism to calculate the formation en-

ergy

Since in our case we are dealing with isolated clusters, hence the boundary

condition that we are using can be called Free Boundary Condition (FBC).

Let us now take a look at the respective expressions, under both the frame-

works of PBC and FBC, to calculate the formation energy of vacancies. In an

overview study of defect analysis Van de Walle[98] et. al. have summarised

a general formula to calculate the formation energy of the defects in PBC.

The formation energy of a defect or impurity X in charge state q is defined

102

Figure 5.4: A schematic representation of the surface to bulk interaction inthe isolated silicon clusters.

as

Ef [Xq] = Etot[Xq]− Etot[bulk]−

∑

i

niµi + q[EF + Ev +∆V ]. (5.1)

Etot[X] is the total energy derived from a supercell calculation with one impu-

rity or defectX in the cell, and Etot[bulk] is the total energy for the equivalent

supercell containing only the bulk species (for example Si, Ga, GaN etc.). ni

indicates the number of atoms of type i (host atoms or impurity atoms) that

have been added to (ni > 0) or removed from (ni < 0) the supercell when

the defect or impurity is created, and the µi are the corresponding chemical

potentials of these species. For the sake of clarification here we can describe

chemical potential as a physical quantity that represents the energy of the

reservoirs with which atoms are being exchanged. EF is the Fermi energy,

103

referenced to the valence-band maximum in the bulk. Due to the choice of

this reference, the users of this method need to explicitly put in the energy

of the bulk valence-band maximum, Ev , in the above mentioned expression

for the formation energy of charged states. A correction term DeltaV is

also needed to be added to the rest of the expression, to align the reference

potential in the defect supercell with that in the bulk. A problem when cal-

culating Ev is that in a supercell approach the defect or impurity strongly

affects the band structure. Therefore one cannot simply use Ev as calculated

in the defect supercell. To solve this problem a two-step procedure is used:

(a) The top of the valence band Ev is calculated in the bulk species by per-

forming a band-structure calculation at the Γ point and (b) an alignment

procedure is used in order to align the electrostatic potentials between the

defect supercell and the bulk. The fact that Ev found for the bulk, (i.e. in

a defect-free supercell) cannot be directly applied to the supercell with a de-

fect can be attributed to the long-range nature of the Coulomb potential and

the periodic boundary conditions inherent in the supercell approach. The

creation of the defect has been reported to give rise to a constant shift in the

potential, and this shift cannot be evaluated from supercell calculations alone

since no absolute reference exists for the electrostatic potential in periodic

structures. One common method is to align the electrostatic potentials by

inspecting the potential in the supercell far from the impurity and aligning

it with the electrostatic potential in the bulk species. This leads to a shift in

the reference level ∆V , which needs to be added to Ev in order to obtain a

more appropriate alignment.

5.2.2 FBC formalism to calculate the formation energy

In the scenario with FBC we try to build a formalism in similar terms. Here

also we calculate the total energies of the nano clusters with and without the

defects. To start with let’s put the formula as the following,

Ef = Etot[defect]− Etot[bulk] + µbulk−atom + qtrpd∆µ+ qsysµ. (5.2)

104

As we have done the calculations with vacancies in bulk silicon let me rewrite

the above equation in terms of bulk silicon for the sake of clarity.

Ef [Si(m− vacancies)] = E[n−m]Si+pH −EnSi+pH +mµSi + qtrpd∆µe + qsysµe,

(5.3)

where Ef [Si(m− vacancy)] is the formation energy of the vacancy (monova-

cancies or divacancies), E[n−m]Si+pH and EnSi+pH are the total energies of the

defect and the perfect system respectively. In the nano cluster there are in

general n Si atoms pacivated by p H atoms at the surface. m is the number

of vacancies created by removing atoms, (1 for mono vacancy and 2 for diva-

cancy). µSi is the chemical potential of bulk silicon which can also be thought

of as the energy of 1 bulk silicon atom. The most important and interesting

factors are the last two factors of the formula. Although the physical im-

portance of the quantity qtrpd will be explained in the following sections yet

it can be helpful to put in an overview of what it is in this context. qtrpd is

the amount of charged trapped near the defect as a consequential effect of

the defect formation. It is a section of the deformed electron density near

the defect and its value depends on the charge of the system. ∆ mue is the

difference of the Fermi energy between the defect and the perfect system due

to the occurence of the defect. qsys is the charge of the entire system for the

charged cases. It is of no use for the uncharged cases. µe is the Fermi level

of a particular charged system which can be considered as the reservoir from

where the charge is taken. To analyse the formula further we have to analyse

the electron density of the different defect systems.

5.3 Density analysis of the defect systems

5.3.1 Uncharged clusters

As it was stated earlier there will be surface effects in the approach within the

FBC formalism. The surface of the nanoclusters can cause some perturbation

in the electron density per unit volume near the center of the cluster. The

reason behind this can be attributed to the presence of the silicon-hydrogen

105

electr

on de

nsity

per u

nit vo

lume

electron density distribution of the perfect cluster

distance from the center in Bohr

surfacecenter

Figure 5.5: The variation of the electron density per unit volume aroundeach shell of the nano cluster (a) for all the clusters (b) for the cluster ofradius 1.2 nm

bonds at the surface which are electronically different from the bulk silicon-

silicon bonds. For our analysis we have studied clusters from 0.6 nm to

1.2 nm. The surface induced phenomenon may indeed bring about a charge

separation between the surface and the core center of the cluster. If we see at

the figure with the variation in the electron density per unit volume around

each shell of the nanocluster of a perfect cluster (without any defect) we

understand that the behavior near the center is similar to that of the PBC

calculations. As it is mentioned in the figure because of the surface and the

surface hydrogens the electron density trails off to zero. In the figure 5.5(a)

all the densities are plotted on the same plot so as to give a comparative idea

of the behavior of the densities in clusters of different sizes. We see from

the figure 5.6(b) that only near the vacancy (di and mono for this case) the

density per unit volume of the shell tends to differ from each other. For the

rest of it the curves are nicely superimposing each other. There is also a

question of the convergence of the observed and calculated physical quantities

with the cluster size. This question will precisely be address in the coming

sections. Let us have a look at the behavior of the density for the defect

states. In the figures 5.6 (a) and (b) it is shown how the electron density per

unit volume near the defect is disturbed for different cluster sizes. Near the

center we see because of the presence of the defects the electronic arrangement

106

Figure 5.6: The variation of the electron density per unit volume aroundeach shell of the nano cluster with a monovacancy (a) for all the clusters. (b)for all the clusters near the vacancy.

is different from that of the perfect clusters. A close look even at the density

distribution of the divacancy clusters would confirm the perturbation of the

density due to the presence of the vacancies. If we look carefully in figure

5.8 with the density plots of the largest cluster with a monovacancy and

divacancy respectively, we will see a clear difference in the electron density

distribution near the vacancies between the monovacancy clusters and the

divacancy clusters. It forces us to think that this might be the effect of

the surface and it triggers the thought of any charge separation between the

surface and the center of the nano cluster.

107

Figure 5.7: The variation of the electron density per unit volume aroundeach shell of the nano cluster with a divacancy (a) for all the clusters. (b)for all the clusters near the vacancy.

5.3.2 Charged clusters

In this section we will have a look around the vacancy in the charged clus-

ters. The main aim of the thesis is to create a formalism with which the

charged systems can be dealt without the image charge/defect interacts of

PBC formalism. Hence it is more interesting here to look into what happens

to the electron density distribution in the nano clusters when the defects

are charged. The overall density plots reveal a fact that near the vacancy

the electronic density is disrupted due to the presence of surface. And the

zoomed in figures also indicate that the charges of the system causes further

108

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 5 10 15 20 25 30

elec

tron

ic d

ensi

ty p

er u

nit v

olum

e

distance from the vacancy in Bohr

electronic distribution of the neutral monovacancy cluster

surfacevacancy

surfacevacancy

1.2nm monovacancy1.2nm divacancy

dBulk

Figure 5.8: The difference in the electron density distribution near the va-cancies between the monovacancy clusters and the divacancy clusters. Thecluster size is 1.2nm

perturbation in the electron density near the vacancy. The question that

one may think about now is what is the effect of the system charge to the

electron density around the vacancy? We see from figures 5.8 , 5.9 and 5.10

of this section that only near the vacancy the density per unit volume of

the shell tends to differ from each other. For the rest of it the curves are

nicely superimposing each other. A further analysis with another tool would

eventually reveal the finer and clearer picture.

Mulliken analysis

Mulliken population analysis[99] is one tool which helps to estimate the par-

tial atomic charges from calculations carried out by few different methods of

109

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 5 10 15 20 25 30

elec

tron

ic d

ensi

ty p

er u

nit v

olum

e

distance from the vacancy in Bohr

electronic distribution of the neutral monovacancy cluster

surfacevacancy

monodBulk1ncv2ncv1pcv

Figure 5.9: The difference in the electron density distribution near the va-cancies for a monovacancy cluster with different charge states.

computational chemistry, particularly those based on the linear combination

of atomic orbitals molecular orbital method. The charge or spin density dis-

tribution is studied as one of the most important properties of a molecule or

a cluster. Although there is no unique definition of how many electrons are

attached to an atom in a molecule or in a cluster it has nevertheless proven

useful in many cases to perform such population analyses. Due to its sim-

plicity the Mulliken population analysis has become a very familiar method

to count electrons to be associated with an atom. As an end result of this

calculation what is obtained is the gross charge in any atomic orbital. Which

is defined in this theory as the difference between the number of electrons

in an atomic orbital and the total number of electrons in the ground state

of the free neutral atom. For our research this analysis is done with the

110

Figure 5.10: The variation of the electron density per unit volume aroundeach shell of the nano cluster with a monovacancy (a) with 1 negative charge(b) with 2 negative charge (c) with 1 positive charge.

Figure 5.11: The variation of the electron density per unit volume aroundeach shell of the nano cluster with a monovacancy (a) with 1 negative charge(b) with 2 negative charge (c) with 1 positive charge , around the vacancy.

nano clusters of all sizes for all charge states and vacancies. The following

results will illustrate the difference in the population of the electrons of the

atoms around the vacancy. From figure 5.12 as one can understand that the

gross charge around each atom is pretty much the same for all the systems

except for the zone around the vacancy. Due to the nature of the bonds deep

inside the bulk we see that the number of electrons in a Si atomic orbital

is slighlty greater than the total number of electrons in the ground state of

the free neutral Si atom. As we go towards the surface due to the presence

of the hydrogens we experience the reverse the phenomenon. Right at the

surface the hydrogens tend to pull the electron density towards themselves

being more electronegative than silicon. Let us see for the fact that indeed

the discrepancy of the electron population is manifested near the vacancy or

at the center of the cluster. In figure 5.13 (a) we show all the gross charge

plots superimposed on each other, and we see that except for the zone near

111

Figure 5.12: The gross charge around the atoms in a cluster of size 1.2 nm for(a) perfect cluster (b) monovacancy cluster with charge -1 (c) monovacancycluster with charge +1 (d) monovacancy cluster with charge -2. The x-axisrepresents the atom ids in general. The hydrogens hence have the latter idsas the clusters are built atom centered.

the center of the cluster, in rest of the zones the plots are almost identical.

At this point it is felt necessary to comment on a small difference that is

observed for the case of the system with charge +1. As far as the hydrogens

are concerned they are more electronegative than Silicon and hence pull the

electron cloud towards themselves. But due to the presence of the positive

charge in the system and, as we will show how it also affects the charge

trapping near the vacancy, there is a Coulombic interaction between the pos-

itive charge of the system and the stretched electron cloud near the surface.

This gives rise to a variation in the values of the grosscharges for the system

with charge +1. This fact forces us to look deeper and closer in the zone of

the mono vacancy (figure 5.13 (b)). Here we see that corresponding to each

112

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 100 200 300 400 500 600

Gro

ss

Ch

arg

ein

e

number of atoms

mono1ncv2ncv1pcv

perfect

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 5 10 15 20 25

Gro

ss

Ch

arg

ein

e

number of atoms

mono1ncv2ncv1pcv

perfect

Figure 5.13: The gross charge around the atoms in a cluster of size 1.2 nmfor (a) all clusters (b) all clusters around the vacancy.

charge state there is a different behavior of the grosscharge for that particular

charge state. Since the 15th atom or the third shell around the vacancy the

gross charges tend to behave similarly. From the above figure it is not easy

to quantify this effect. Now to quantify this effect we use the expression,

Qtrpd = Grosschargedefect −Grosschargeperfect, (5.4)

where Qtrpd can be thought of a residual charge which is trapped near the

vacancy, which is the difference between the grosscharge of the defect system

of any charge state to that of the perfect system. Now we can try to find out

whether this Qtrpd can be considered physical for a given analysis. Indeed if

we can see that this quantity is a converged one with the size of the nano

113

clusters then we can understand the significance of this physical quantity in

a broader physical sense. In figure 5.14 we see that with the cluster size

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

Tra

pped

Charg

ein

e

mono1ncv1pcv2ncv

Radius of the nanocluster in nm

Figure 5.14: The convergenge of Qtrpd with cluster size.

the quantity Qtrpd converges to values for a cluster for each of the different

charge states. We can see from the figure that the values of the Qtrpd have

converged since the cluster size of 1.0nm, and it is only natural to consider

clusters from 1.0nm for further calculation of physical quantities. A striking

fact is that even if the value is very low yet the uncharged mono vacancy is

a bit negatively charged. And we can see that increase in the charge of the

system increases the value of Qtrpd. Hence Qtrpd is the most negative for the

most negatively charged system. This of course gives an impression of the

behavior of vacancies in a charged nano cluster. We can conclude from here

that indeed the defect in the nano clusters has a tendency to trap charges

and that quantity is dependent of the charge of the system.

114

Physical origin of the trapped charge

The presence of the trapped charges near the vacancy in the nano clusters

is a physical phenomenon which must have a relation any change in other

physical properties of the system. Results have shown as cited earlier that

presence of defects changes the HOMO-LUMO gap and the band structure

of any material, silicon for example in this thesis. Since we are dealing

with FBC we cannot do a band structure calculation, but a diagonalisation

method to calculate the HOMO-LUMO gap for the nano clusters can be used

instead. In this mode of calculation we calculate the HOMO-LUMO gaps

for the nano clusters of all sizes and charge states i.e. the difference between

the highest occupied molecular orbital(HOMO) and the lowest unoccupied

molecular orbital(LUMO). In the Free Boundary Conditions the HOMO-

LUMO gap is found to be heavily overestimated for smaller clusters. But

the HOMO-LUMO gap of the larger clusters are found to be converging

close to the bulk value. There are questions of quantum confinement in

the nano clusters of course. And the fact the HOMO-LUMO gap is heavily

overesmitated in very small nano clusters owe its origin to this confinement

effect. In figure 5.15 we have aligned HOMO levels so as to understand

the effect of the presence of the defects. The extreme-right column shows

the HOMO-LUMO gap as calculated from the 1.2 nm perfect cluster. The

middle column and the extreme left column are for the neutral monovacancy

and divacancy respectively. There we can see that indeed there are states

introduced within the HOMO-LUMO gap of the perfect nano clusters. Which

means creating one defect might mean otherwise to introduce energy states

within the HOMO-LUMO gap of a perfect system. We can investigate more

in the same spirit the system of the charged monovacancy. In figure 5.16 we

have shown a similar figure. Here we see that taking away one electron from

the system or giving one more electron to the system reduces the HOMO-

LUMO gap. In other words this can be though of as an effective way to

reduce the barrier of the electron flow in the bulk silicon. But there is

something which is found to be striking in this analysis. It seems that an

excess of negative charge in the nanoclusters increases the HOMO-LUMO gap

115

Figure 5.15: Representiing the change in the HOMO-LUMO gap due tothe presence of defects (uncharged) (-1 means mono vacancy and -2 meansdivacancy)

.

again. In reality the HOMO-LUMO gap of a neutral monovacancy cluster is

narrower than a monovacancy cluster of charge -2. We have already seen with

the mulliken analysis plot that increase in the negative charge of the system

will increase the negativity of the charge at the center. It may be because of

this charge an excess of charge is not welcome due to the coulombic repulsion

between the excess charge and the trapped charges. The argument is still an

open matter for further analysis and explanation. Now coming back to the

origin of the trapped charges again we must take into consideration the fact

that due to the presence of the defects and the charges there is a shift in the

chemical potential of the respective charge-defect systems. This shift can be

quantified as

∆µe = µeperfect − µedefect . (5.5)

116

Figure 5.16: Representiing the change in the HOMO-LUMO gap due to thepresence of defects (charged monovacancy with charges = -2,-1,0,1)

.

And the trapped charge Qtrpd is related to this shift in chemical potential

by the energy required for that charge to be brought at the center around

the vacancy in the cluster and we can quantify that energy as

Etrappedcharge = Qtrpd∆µe. (5.6)

This expression thus arrives in the formula of calculation of the formation

energy. Now the following motivation would be to look for an analytical

electrostatic model which must simulate this sort of a charge separation in

the nanoclusters.

5.4 Comparison of results obtained

Table 5.1 shows the performance of the model we have used to calculate the

formation energies. Experimental values in the spirit of the work by Watkins

117

Table 5.1: Table representing the comparison of the values of the formationenergies of monovacancies as calculated under PBC[75] and FBC(this work)and experiment[3].

Charge PBC FBC Expt.0 3.605 3.70 3.602-1 4.319 4.63 —-2 4.909 4.67 —1 3.545 3.86 —

and Corbett are still not available for the charged monovacancies and hence

it is still a point of time to talk about the accuracy of the model in discussion.

Yet we see our results are not far from the PBC values. Here some things

are to be kept in mind. The important thing is the fact that our motive

is to create a model without defect-defect interaction and devoid of any

background charge concept. In the following sections we will put forward an

analytical model to simulate the electrostatics of the given FBC formalism

which would indeed prove that our results have a very good chance to be

accurate with respect to the future experiments. The question of convergence

can be raised here and we will show in due course of this thesis that the

convergence issue can be tackled well with the electrostatic and elastic model

of the entire formalism.

5.5 Electrostatic model

In this work we have worked out an electrostatic scheme which appears fit

the atomistic model representation of the defects in bulk silicon and helps

to incorporate the concept of the trapped charges into an analytical model.

As the following figure will show in this approach we deal with an isolated

system free from all the interactions between the image charges and the real

bulk defect. Of course we have a surface to bulk interaction but as it will be

shown in the following analysis that this surface to bulk interaction indeed

helps to picturise the real electrostatic scenario of the cluster approach. In

the following section we will try to understand how this gives rise to the

118

Figure 5.17: Spherical shell of radius R with a smeared surface charge and anet charge at the center with the tiny sphere.

previously shown phenomenon of trapped charges.

5.5.1 Overview

Since the surface electron density interacts with the somewhat perturbed

electron density around the defect, it can be apprehended that this interac-

tion will cause some charge separation between the surface and the center

of the cluster. In this analysis we are dealing with the vacancies. Because

of the absence of the atoms there are dangling bonds from the neighboring

atoms of the vacancies. These electron density interacts with the surface and

as a consequence of the shift in chemical porential we find the occurence of

the trapped charges. The electrostatics of the entire system can be thought

of a problem where it is the objective to find out the elctrostatic energy that

can be stored in a cluster of ’n’ Silicon atoms passivated by ’p’ Hydrogen

atoms at the surface. The first model that is considered here is for that of

an uncharged cluster of radius ’r’. Although in reality the clusters are not

strictly spherical they are assumed to be perfect spheres in the model.

119

5.5.2 Details

Well as shown in figure 5.17 let the cluster be a spherical shell of radius R

with a net positive charge q placed at the center and a net positive chage −q

smeared all over the surface of the spherical shell. The spherical shell is filled

with a dielectric substance with a dielectric constant ǫr. Among the various

methods used to calculate the total electrostatic energy stored in any system,

the one that is used here is with the method of finding the electrostatic field

in different regions of the system. Then the electrostatic energy stored in the

system will be given by:

W =ǫ02

∫ǫrE

2 dτ (5.7)

where the symbols have usual meaning.

Let us divide the regions in our system into two different parts, viz. for the

electrostatic field inside the spherical shell, EI , and for the electrostatic field

outside the spherical shell, EII . If we consider a Gaussian surface outside

the spherical shell we can readily see that the total charge enclosed within

the Gaussian surface is 0 and hence the field outside the spherical shell, EII ,

is evidently 0.

Now to calculate the field inside the spherical shell we consider a sphericall

Gaussian surface along the surface of the spherical shell of our model. Here

the total charge enclosed is q. If we apply Gauss Law here we get:

∫EI . da =

Qenclosed

ǫ=

q

ǫ0ǫr(5.8)

By symmetry considerations we can see that the field inside is constant and

is radially outward. Hence we can easily deduce that the field inside the shell

is:

120

EI =q

4πǫ0ǫrr2(5.9)

Now to calculate the total energy stored in the system we fall back to equa-

tion (1) and integrate over the spherical volume element dτ = r2Sinθdθdφdr.

Hence the final expression of the total electrostatic energy stored in the sys-

tem is given by:

W =ǫ02

∫ǫr(

q

4πǫ0ǫrr2)2r2Sinθdθdφdr (5.10)

where the integrating limits for θ and φ are from 0 to π and 0 to 2π re-

spectively and hence integrating over θ and φ we get 4π. And the limit of

integration for r is from 0 to R. Hence from equation (4) we get:

W =q2

8πǫ0ǫr

∫ R

0

1

r2dr (5.11)

5.5.3 Modification

From equation (5.11) we can see that there is a problem, for the integrated

value at r = 0 is definitely blowing up. And it is evident from the fact that

the electrostatic energy of a point charge is indeed infinite. Hence we feel

the need of modifying our model a bit. Let us now assume another very tiny

spherical shell around the charge q. And we restrict ourselves in knowing the

electrostatic energy within the region between the two shells. Let the radius

of the tiny shell be R1. Now equation (5.11) becomes integrable again. The

total energy stored within the region of our interest is now given by:

W =q2

8πǫ0ǫr[1

R1

− 1

R] (5.12)

121

5.5.4 Remarks

The units of the quantities in the final expression of the total electrostatic

energy are all in S.I. units. Hence the energy that will be computed from this

expression will be in Joules(J). To compare with the total electronic energies

of our systems we need to convert this energy in to electron-Volts (eV). The

importance of this model will be explained in the following sections as we will

see the correspondence of the functional similarity of derived expression to

the actually calculated values. We wish to see whether this analytical model

is suitable to simulate the real electrostatics of the system. In the section

where we discuss the validity of this model we will show that the convergence

of the formation energy of the vacancies follow the same functional trajectory

as the electrostatics mentioned in this section.

5.6 Geometry of defects in nanoclusters

An important section of study regarding the defects is their geometry. A

defect essentially is an inconsistency in the geometrical structure whose pres-

ence will alter the electronic structure of the material. The change in the

electronic structure is often manifested in the geometrical changes that take

place due to the defects. The movement of atoms around a vacancy or a

foreign substitutional element can be thought of as a primary example. In a

report[100] D.V. Makhov and L.J. Lewis have talked about the distortions for

the divacancy in silicon. In another letter[101] S. Ogut and J. R. Chelikowsky

have reported similar distorsions around the divacancies in crystalline silicon.

The general understanding of these distorsions is based upon an electronic

phenomenon known as the Jahn-Teller distorsion.

5.6.1 Jahn-Teller Distorsion (JT)

The formal statement of the Jahn-Teller theorem states that in a nonlin-

ear molecule, if degenerate orbitals are asymmetrically occupied, a distortion

will occur to remove the degeneracy or in other words in an electronically

degenerate state, a nonlinear molecule undergoes distortion to remove the

122

degeneracy by lowering the symmetry and thus by lowering the energy. In

context of the study of defects it is found that due to the removal of an atom

to create a vacancy (or more) an untended degeneracy is enforced upon the

system. To remove that, the atoms around the vacancy would move in such a

fashion which would cause the total energy of the system to go down in value

to attain a more stable state. Hence the real manifestation of this electronic

effect of the removal of the degeneracy is found in the change in the geometry

of the system in and around the defect. Since the pioneering electron param-

agnetic resonance (EPR) studies of Watkins and Corbett[3], there have been

controversies about the electronic and atomic structures of the divacancy in

crystalline Si. These controversies have been concerned with the exact nature

of the above mentioned symmetry-lowering Jahn-Teller distortions that split

the degenerate deep levels. There are of course many reports which would

reveal the fact that the sense and the magnitude of these distortions, as in-

ferred from the EPR data and theoretical calculations, have been at variance.

There can be two different manifestations of these JT distorsion. The reso-

nant bond (RB) and the large pairing (LP) distorsions. In the study of D.V.

Makhov and L.J. Lewis[100] a schematic diagram (figure 5.18) shown the

difference between these two types. Although the results of Ogut and Che-

likowsky matched the experimental analysis of Watkins and Corbett there

were reports based on the results of ab initio calculations for divacancies by

Saito and Oshiyama[102] proposed the inverse distortion mode, the RB mode.

In the calculation as reported in works of D.V. Makhov and L.J. Lewis[100],

they found that the resonant bond distortion has slightly lower energy than

the pairing distortion ( 10 meV). And they also mentioned that the transi-

tion from one mode to the other occurs without any potential barrier and is

accompanied by a rotation of the symmetry plane of the divacancy. Hence

they claim that at room temperature, a divacancy should oscillate between

these two distortion modes.

123

Figure 5.18: Schematic view of LP and RB divacancy distortion modes..

5.6.2 Jahn-Teller Distorsion in the uncharged nanoclus-

ters

In their work[97] Ogut and his colleagues have studied the size dependence

of the relaxations of the atoms around the vacancy with respect to the shells

around the vacancy. Their findings were such which proved the inward move-

ment of the neighboring atoms towards the vacancy. They even found the

same for the charged cases. In the study that we undertook we also tried to

find whether it is possible to observe such distortions in these silicon nan-

oclusters. We have tried to investigate the geometry of the vacancies in

similar spirit. We did full geometry relaxation of the atoms and the cutoff

for the maximum value of force in all direction on each atom was held at

10−5Hartree/Bohr which is equivalent to 0.0005eV/A. In our analysis with

the neutral divacancies we found that depending on the size of the clusters

RB and LP distorsions take place around the vacancy. These distortions are

therefore dependent in a particular fashion on the cluster sizes and a general

statement relating the type of distorsion to that of the size of cluster can be

124

made. In the figure 5.19 we demonstrate the nearest neighbors of the neutral

divacancy of clusters 0.9 nm and 1.2 nm. In this figure as we see the nan-

Figure 5.19: JT distorsions and the distance between the nearest neighborsof the neutral divacancy for (a) 0.9 nm cluster and (b) 1.2 nm cluster

ocluster of radius 0.9 nm exhibits the nearest meighbours of the divacancy

in a LP distorsion yet the 1.2 nm cluster divacancy has undergone a RB

distorsion. The clusters of sizes more 0.9 nm upto 1.2 nm have all exhibitted

the same tendency. This fact can be addressed with a careful observation

to the surface effect of the nanoclusters. Although the nanoclusters are con-

sidered spherical in the electrostatic model yet they are not spherical and

they have the planes of the polygons on each face. The arrangement of the

planes are different in clusters of different size and hence the surface effect

is not homogeneous or consistent in nature for all the clusters. And we may

be forced to think that this effect in the nanoclusters owe its origin to the

arbitrariness of the arrangement of the surfaces. To understand the effect of

the surfaces let us have a look at the monovacancies, figure 5.20. Here also

we can see the JT effect around the monovacancy. If we look carefully we

will notice the movement of the nearest neighbors to the vacancy in the two

clusters appears different even if viewed from the same direction. By this

I mean the movement of the neighbors in a preferred direction. Given the

isotropy of an infinite bulk this indeed is not an issue. One can argue that

in a bulk (which we are trying to simulate through our nanoclusters) these

125

Figure 5.20: JT distorsions and the distance between the nearest neighborsof the neutral mono vacancy for (a) 1.1 nm cluster and (b) 1.2 nm cluster

two apparently different movements are equivalent. But for our system as it

is finite the movement of the atoms are of course influenced by the surfaces

present in a particular cluster. A schematic diagram (figure 5.21) will be

helpful to understand this fact. For the sake of completeness here it must be

Figure 5.21: The two modes of JT distorsions as observed in the nanoclus-ters. When viewed from a particular direction in a finite system they appeardifferent of course. The box inside is the vacancy and the round balls are theneighboring atoms.

mentioned that out of these two different movements both are equally likely

as it does not depend on the cluster size as it was for the divacancies.

126

Table 5.2: Table representing the extent of the deformations in the distancebetween the nearest neighbors of the monovacancy in different charge states.Avg.4 is the column displaying the average distance between the 4 neighborswhich are at same distance from the vacancy and Avg.2 is for the averagedistance between the 2 neighbors which are at the same distance from thevacancy.

Charge Avg.4 Avg.2 % Deformation0 3.572 2.996 -16.13-1 3.464 3.058 -11.72+1 3.8 3.556 -6.42-2 3.094 3.423 +10.63

5.6.3 Jahn-Teller Distorsion in the charged nanoclus-

ters

Let us now investigate the extent of this JT distorsion in the charged systems.

As we have already seen that the presence of charge in the system affects

the electron density near the vacancy it is quite natural to assume that the

effect will be seen in the geometry of the system as well. In fact, as it is

already mentioned before, the geometry is the manifestation of the electronic

changes in the system due to the presence of the defects. Let us have a

look at the comparative JT distorsions in the neutral and variously charged

monovacancy clusters.

Table 5.2 along with figure 5.22 exhibits a very strange fact. Indeed from

figure 5.22 we understand that the presence of charge in the system and

due to the effect of the trapped charges near the vacancy the geometrical

manifestation of the JT distortion is not the same in all the charged clusters.

In fact we see that for all the clusters there is a tendency of 2 atoms moving

such that they share a similar distance from the vacancy and their distances

from the other atoms are larger of course. This is also shown in the schematic

figure as mentioned before in context of the two types of distortions in the

finite clusters. And we can have a measure of the deformation from the above

table. The table shows that for charge states 0, -1 and +1 the distance

between the atoms grouping together is respectively 16.13, 11.72 and 6.42

127

Figure 5.22: JT distorsions and the distance between the nearest neighborsof the monovacancy for 1.2 nm cluster with charges (a) 0 (b)-1 (c)1 (d)-2

percent less than the atoms being apart. Where as for the charge state -2

the distance between the atoms grouping together is 10.63 percent more than

that of the other atoms which have moved apart. In this regard we may recall

that even from the HOMO-LUMO gap analysis in the previous section we

have known the system with -2 charge behaves differently from the others.

This different mode of JT distortion must therefore be caused due to the

effect of the trapped electron density near the center of the nanocluster.

128

5.7 A simple force model to simulate the JT

distorsion

Figure 5.23: Schematic representation of a simple force model of JT distorsion.

Owing to the fact that the atoms are moving as a manifestation of the

JT distortion a simple force model can be visualized in accordance with

the overall picture. In the schematic figure (figure 5.23) of the model the

inward movement of the atoms towards the center of the cluster (around the

vacancy) is represented. Due to the movement of the atoms the volume of

the defect is changed by dV . The forces exerted on this defect volume are

from all possible directions. And hence we can imagine a contour of circular

areas as shown in the figure. The force per unit area for each such circular

area is the pressure P exerted on the volume dV and the work done to create

dV would be given as the elastic energy,

Welastic = PdV. (5.13)

and

P =Force

Area=

C

r2(5.14)

where C is a constant and r is the radius of the nanocluster.

129

5.8 The combination of the electrostatic and

simple force model

If we fall back on the electrostatic energy stored in the cluster to bring all

the charges together the general form of the expression would look like,

Welectrostatic = A− B

r. (5.15)

Hence if we combine the two models, i.e. electrostatic model and the simple

force model of JT distorsion we will get the total energy required to form the

vacancies in a nanocluster with the JT distorsion. Therefore we can say that

the total energy required to form the vacancies in a nanocluster with the JT

distorsion must have the analytical form of,

Eformation = A− B

r+

C

r2.(5.16)

Now we can check the validity of this functional form with different values

of the radii of the nanoclusters and then can simultaneouly plot the values

of the calculated formation energies under the FBC formalism. This as will

be explained in the through following figures will also help us to understand

the convergence of the value of the formation energy in relation with the

cluster size. As it is evident from the fitted curve (figure 5.24) that the

value of the formation energy seems to be converging with the cluster size

and hence the values quoted from the calculations of the 1.2 nm clusters can

be considered to be accurate under the given formalism. This curve also

indicates the physically acceptable nature of the models discussed above.

The points which are away from the analytical curve are of course results of

the finite size effect as one must experience in calculations like these. It also

emphasies the presence of a strong surface effect when the clusters are really

small. Hence it can be argued that the physical significance of the formation

energy as calculated from the cluster of size 1.2 nm can be taken as standards

for future bulk calculations.

130

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

form

atio

n en

ergy

in e

V

radius of nanoclusters in nm

A - B/r +C/r2calculated plot

Figure 5.24: The comparison of the calculated formation energies with theanalytical model and the convergence of the formation energy value withrespect to the cluster size

.

5.9 Conclusions

As concluding remarks the following observations can be considered as the

result and immediate finidings of the research.

• The electrostatics of the nanoclusters with defect is relatively simple

and easy as compared to that of the PBC systems. The requirement

of a background compensating charge is not required.

• It is also noteworthy that the defect-defect interaction along with in-

teraction of the image charges can be avoided for the study of charge

systems. Of course there is a surface to bulk effect which is an artefact

of the FBC model but it is relatively simpler to deal with in the con-

text of finding the formation energies of the defects and studying there

geometries.

131

• The geometry of the charged defects in nanoclusters manifests most the

effect of the surface. This is a bit far stretched one may presume. But

indeed the surface induces the occurrence of trapped charges near the

vacancy. And we have show that the geometry for each charged state

is different from one another and it is obviously due to the different

values of the trapped charges. Hence we see different deformations in

the JT representations for different charge states. Also the forms of

distortion like the RB and LP distortions are observed in this kind of

an approach.

• The phenomenon of trapped charges is observed as an important feature

of the FBC model which allowed us to understand the electrostatics of

the nanoclusters with or without defects. We may even emphasize the

ability of the vacancies to trap charges in a solid as we have found

and proved in our analyses. Even for the case of PBC we have seen if

we somehow manage to keep a charge in the system far enough from

the vacancy (one cannot actually be very far from the vacancy in PBC

because of the periodicity itself) it tends to trap that charge. Hence in

general one may infer that vacancies trap charges.

5.10 Future perspectives

5.10.1 Different other defects

This analysis can be extended to the study of other defects as well. In fact

even within this research we have tried to get similar results for the charged

di vacancies as well. But unfortunately the required criteria for an acceptable

value of forces on the atoms are not reached. The reasons may be hidden

in the behavior of trapped charges for the divacancies. The electron density

around the divacancy must be disturbed in a greater extent due to the surface

and a spin polarized calculation of the nanoclusters would be preferred to

look into the matter more deeply. All the calculations up to now are spin

average calculations. But to do a spin polarized calculation on a cluster of

132

1.2 nm in BigDFT, the expense is huge and hence it is a bit difficult to tackle

at the moment.

5.10.2 Migration energies

Another method to verify whether the results obtained by this method are

believable or not is to check for other quantities which are reported in experi-

ments. One such method is to calculate the migration energy of a vacancy. In

figure 5.26 we can see a migration of vacancy pictorially. In reality we have

Intermediate state

Energy we need

Figure 5.25: Scematic diagram to explain the concept of the migration energy.

tried already to calculate the migration energy of monovacancy within the

framework of our model. We have used the DIIS[103] method to find the mid-

dle point of the journey of the vacancy from one place to the other. As per

our calculations we have found out that the migration energy of the neutral

monovacancy in 0.5 eV. As reported in the report by Watkins[63] the experi-

mental value of the migration energy of the neutral monovacancy is 0.45 eV.

This result encourages us to continue to do the same for other charge vacan-

cies. Indeed along with these calculations another form of calculation is still

under process. To get the contour of the entire path of migration a common

procedure is to do a Nudged Elastic Band (NEB) calculation. Where all the

intermediate replicas are created and total energy calculations are done on

each replica. This calculation will help us to know the entire trajectory of

the vacancies. But this is also to be kept in mind that these calculations are

very expensive in time.

133

xyz

vacancy

xyz

movementof the vacancy

Figure 5.26: The real picture of the movement of the vacancy

5.10.3 Homothetic clusters

There is a possibility to account for the surface effect in a more analytical

and quantitative manner, if we can create the clusters of different sizes with

the same family of surfaces. Clusters of different sizes with same surfaces

are called homothetic clusters. If we can put one cluster inside the other one

they will make something like concentric clusters. We have already succeeded

in making homothetic clusters of three different sizes but calculations with

them took us out of the time limit of the present thesis. We believe similar

calculations as done within the framework of this thesis will yield interesting

quantitative facts about th surface effect in the nanoclusters.

134

xy

z

xy

z

xy

z

xy

z

Figure 5.27: Homothetic clusters made from boxes of (a)64 (b)216 (c)512(d)1000 silicon atoms.

135

Chapter 6

Final words and Perspectives

In the quest to study the charged defects of silicon we have managed to

touch upon and dig into many different related topics in this branch of con-

densed matter physics. In terms of field study, we have gathered knowledge

about almost all the techniques in the DFT framework, that scientists use to

calculate various physical quantities regarding the charged defects. In that

kind of a study we have summarized the advantages and disadvantages of

each method. Indeed PBC methods are good to simulate a bulk, but the

problem of image defect interactions is something which we cannot get rid of

anyhow, barring the use of some correction factors. In the main part of this

research we have tried to find an approach which would be different from the

PBC formalisms. We have felt the need of not using the background com-

pensating charge in the PBC formalism for the calculation of the charged

defects. The backbone of this research is the DFT method as all the total

energy calculations are done with the wavelet-based DFT code - BigDFT.

Hence a brief yet detailed description of the DFT framework is included in

the thesis. The important concepts about the basis sets, pseuopotentials and

the exchange correlation functionals are also discussed in here. In the study

with the HGH pseudopotentials, we have tried to show the reliability of those

pseudos with the inclusion of the NLCC. We have shown of course that with

the NLCC implemented within the HGH pseudopotentials, we can use them

for very accurate calculations, as comparable to the all electron calculations.

136

This is a major step towards the pseudopotential calculations as it takes us

towards the ultimate aim, which is to use a complexity-less scheme for accu-

rate research. The main part of this thesis as I have already mentioned was

to find a novel way to calculate the formation energy of the charged defects.

We have used nanoclusters to simulate the defects and have found that the

formation energy thus found are in accordance with whatever experimental

values we have. We have put forward the phenomenon of the trapped charges

near the vacancy in the nanoclusters which we believe affects the geometry

and the energy states of the defect. Keeping that in mind we have given here

an analytical electrostatic model to simulate the charge distribution in the

nanoclusters. We have also done various numerical analyses to provide ample

proof for the electrostatic model of the system. Although an admitted fact is

that the PBC method employs less atoms than our FBC method but in our

method we have dealt away with the correction factors needed otherwise for

the PBC method. In future these analyses with the correct electrostatics can

be used for other systems and of course for various other studies like the mi-

gration energy and diffusion of defects. We can study the effect of charge in

the substitutional and interstitial defects in silicon with this cluster method.

In principle the success and validity of this method would ensure further

application of this to study all other point defects in this new method. It

would be very useful to know about the energetics and the geometry of such

defects in nanoclusters. There is of course a lot of room to study magnetic

defects as well. Nanoclusters which are magnetic can also be studies within

similar formalism. The effect of the trapped charges and the surfaces in the

magnetic systems can be extremely interesting to look for. We can look for

the various effects of the dilute doping defects and the resultant magnetism in

the metal oxide systems with this method. This method will provide us with

an alternative way to the model more frequently used hamiltonian methods.

In a nutshell we can say a new method to study the energetics of the defect

will open up channels of research with defects in all possible environment.

With the recent development to include the applicability of the Wannier

functions in BigDFT, we can be hopeful of using this cluster approach to

137

carry out somewhat an order-N method, in terms of localization regions,

to study some embedded systems. As for example we can talk about the

molecules on surfaces which are semi-infinite systems, or even the charged

defects in bulk(infinite systems). The correct electrostatics of the cluster ap-

proach makes it transferable to the embedded system calculations. Since the

wavelets are localized, for each orbital a region of localization can be defined

. All the operations can be carried out as applications of the Hamiltonian

in the given orbital. Hence methods and theoretical tools can be developed

based on order N method to perform calculation of an atomic system embed-

ded by a tight-binding model based on Wannier functions. If the Wannier

orbitals of Si bulk can be calculated beforehand, then, using tight binding

methods on Wannier, one can define a hamiltonian and a self-consistent field

to calculate an infinite system of Si bulk. Following steps would include the

processes to deal with the defect in the embedded environment along with

the relaxation of the Wannier orbitals in a cluster centered around a vacancy.

It will be feasible with this mixed scheme (wavelet + tight-bindings) to de-

scribe properly charged trapped states even if dimension of the system exceed

many nanometers. It is worth mentioning here that wavelets being localized

themselves make it possible to do locally all operations as applications of the

Hamiltonian. It terms of technical complexities wavelets are advantageous

than plane waves in this particular aspect. Hence we can consider the clus-

ter approach, as we have described in the larger part of this thesis, as the

first step in the field of an order-N method to study the embedded systems.

With the electrostatics and the energetics known for the cluster method the

following research procedures will be immensely facilitated .

138

Bibliography

[1] D. A. Faux, J. R. Downes, E. P. O’Reilly, J. Appl. Phys., 82, 3754–3762

(1997).

[2] G. A. Baraff, E. O. Kane, M. Schluter, Phys. Rev. Lett., 43, 956–959

(1979).

[3] G. D. Watkins, J. W. Corbett, Phys. Rev., 138, A543–A555 (1965).

[4] G. D. Watkins, J. R. Troxell, Phys. Rev. Lett., 44, 593–596 (1980).

[5] J. C. Bourgoin, J. W. Corbett, Rad. Effects, 36, 157–188 (1978).

[6] R. J. Needs, J. Phys. Condens. Matter 11, 10437–10450 (1999).

[7] W.-K. Leung, R. J. Needs, G. Rajagopal, S. Itoh, S. Ihara, Phys. Rev.

Lett., 83, 2351–2354 (1999).

[8] S. J. Clark, G. J. Ackland, Phys. Rev. B, 56, 47–50 (1997).

[9] V. V. Lukianitsa, Semiconductors, 37, 404–413 (2003).

[10] G. M. Lopez, V. Fiorentini, Phys. Rev. B, 69, 155206–155213 (2004).

[11] A. H. Kalma, J. C. Corelli, Phys. Rev., 173, 734–745 (1968).

[12] L. J. Cheng, J. C. Corelli, J. W. Corbett, G. D. Watkins, Phys. Rev.,

152, 761–774 (1966).

[13] S. Goedecker, T. Deutsch, L. Billard, Phys. Rev. Lett., 88,

235501–235504 (2002).

139

[14] I. Lazanu, S. Lazanu, Phys. Scr., 74, 201–207 (2006).

[15] D.Caliste and P.Pochet, Phys. Rev. Lett. 97, 135901 (2006).

[16] D.Caliste, K.Z.Rushchanskii, and P.Pochet , Appl. Phys. Lett. 98,

031908 (2011).

[17] P. Hohenberg and W. Kohn, Phys. Rev. 136. B864 (1964).

[18] W. Kohn, Reviews of Modern Physics 71 nb. 5 (1999).

[19] W. Kohn and L. J. Sham, Phys. Rev 140, A1133 (1965).

[20] C.-O. Almbladh, U. von Barth Phys., Rev. B 31, nb. 6 (1984).

[21] J. Chen, J. B. Krieger, Y. Li, J.G. Iafrate, Phys. Rev. A 54 nb. 5 (1996).

[22] L Fan, T.J. Ziegler J. Chem. Phys. 95, 7401 (1991).

[23] J. P. Perdew, A. Zunger, Phys. Rev. B 23, nb. 10 (1981).

[24] S. Kurth, J. P. Perdew, Phys. Rev. B 59 nb. 16 (1999)

[25] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865

(1996).

[26] M. Ernzerhof and G. E. Scuseria, J. Chem. Phys. 110, 5029 (1999).

[27] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).

[28] J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys. Rev. Lett. 82,

2544 (1999).

[29] von Barth, U., Physica Scripta. Vol. T109, 9–39, (2004).

[30] Muller, J. E., Jones, R. O. and Harris, J., J. Chem. Phys. 79,

1874(1983).

[31] Terakura, K., Oguchi, T., Williams, A. R. and Kubler, J., Phys. Rev. B

30, 4734 (1984).

140

[32] Wang, C. S., Klein, B. M. and Krakauer, H., Phys. Rev. Lett. 54, 1852

(1985).

[33] Godby, R. W. and Needs, R. J., Phys. Rev. Lett. 62, 1169 (1989).

[34] Parr, R. G. and Yang, W., Density-Functional Theory of Atoms and

Molecules,(Oxford University Press, New York 1989)

[35] von Barth, U., Chem. Scripta 26, 449 (1986).

[36] J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev.

Lett. 91, 146401 (2003).

[37] J. P. Perdew, J. Tao, V. N. Staroverov, and G. E. Scuseria, J. Chem.

Phys. 120, 6898 (2004).

[38] J. Harris and R. O. Jones, J. Phys. F 4, 1170 (1974).

[39] A. Gorling and M. Levy, J. Chem. Phys. 106, 2675 (1997).

[40] A. D. Becke, J. Chem. Phys. 98, 5648 (1998).

[41] E.R.Batista, J.Heyd, R.G.Hennig, B.P.Uberuaga, R.L.Martin,

G.E.Scuseria, C.J.Umrigar and J.W.Wilkins Phys. Rev.B 74, 121102

R(2006).

[42] J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207

(2003); J. Heyd and G. E. Scuseria, ibid. 120, 7274[U+0351](2004) ; J.

Heyd and G. E. Scuseria, ibid. 121, 1187 (2004).

[43] L. Genovese, A. Neelov, S. Goedecker, T. Deutsch, S. Ghasemi, A. Wil-

land, D. Caliste, O. Zilberberg, M. Rayson, A. Bergman,and R. Schneider

, J. Chem. Phys. 129, 014109 2008

[44] D.R. Hamann, M. Schluter, and C. Chiang, Phys. Rev. Lett. 43, 1494

(1979).

[45] G.B. Bachelet, D.R. Hamann, and M. Schluter, Phys. Rev. B 26, 4199

(1982).

141

[46] G. Kerker, J. Phys. C 13, L189 (1980).

[47] D. Vanderbilt, Phys. Rev. B 32, 8412 (1985).

[48] N. Troullier and J.L. Martins, Phys. Rev. B 43, 1993 (1991).

[49] S. Goedecker, M. Teter, and J. Hutter, Phys. Rev. B 54, 1703 (1996).

[50] S. Goedecker , K. Maschke Phys. Rev. A 45, 88–93 (1992) .

[51] C. Hartwigsen, S. Goedecker, and J. Hutter, Phys. Rev. B 58, 3641

(1998).

[52] A.M. Rappe, K.M. Rabe, E. Kaxiras, and J.D. Joannopoulos, Phys.

Rev. B 41, 1227(1990).

[53] L. Kleinman and D.M. Bylander, Phys. Rev. Lett. 48, 1425 (1982).

[54] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).

[55] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, Phys.

Rev. B 47,10142 (1993).

[56] G.P. Francis, and M. C. Payne, J. Phys. Condens. Matter 17, 1643,

(1990).

[57] T. L. Beck, Rev. Mod. Phys. 72, 1041 2000.

[58] J. E. Pask, B. M. Klein, C. Y. Fong, and P. A. Sterne, Phys. Rev. B

59,12352 1999.

[59] J. J. Mortensen, L. B. Hansen, and K. W. Jacobsen, Phys. Rev. B

71,035109 2005.

[60] M.C.Payne, M.P.Teter,D.C.Allan,T.A.Arias,and J.D.Joannopoulos,

Rev. Mod. Phys. 64, 1045(1992).

[61] A.M.Rappe,J.D.Joannopoulos, and P.A.Bash, J. Am. Chem. Soc. 114,

6466(1992).

142

[62] M.Leslie and M.J.Gillian, J. Phys. C 18, 973 (1985).

[63] G. D. Watkins Material science in semiconductor processing 3 227-235

(2000)

[64] G. Feher Phys. Rev. 103 (3): 834–835, (1956)

[65] R. O. Simmons and R. W. Balluffi Phys. Rev. 117, 52–61 (1960).

[66] G.Makov and M.C.Payne, Phys. Rev. B. 51, 7(1995).

[67] S.W.de Leeuw,J.W¿Perram, and E.R.Smith, Proc. R. Soc. London Ser.

A373, 27 (1980).

[68] R.M.Martin, Phys. Rev. B 9,1998 (1974).

[69] Peter A.Schultz, Phys. rev. Let. 84, 9 (2000).

[70] Peter A.Schultz, Phys. rev. B 60, 1551 (1999).

[71] M.Leslie and M.J.Gillian, J. Phys. C 18, 973 (1985).

[72] C.Freysoldt, J.Neugebauer,C.G.Van de Walle, Phys. rev. Let. 102,

016402 (2009).

[73] C. W. M. Castleton, A. Hoglund, and S. Mirbt, Phys. Rev.B 73, 035215

(2006).

[74] J. Shim, E.K. Lee, Y. J. Lee, and R. M. Nieminen, Phys. Rev. B 71,

035206 (2005).

[75] A. F. Wright and N. A. Modine, Phys. Rev. B 74, 235209 (2006).

[76] I. Dabo, B.Kozinsky, N.E.Singh-Miller, and N.Marzari, Phys. Rev. B

77, 115139 (2008)

[77] L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem.

Phys. 106, 1063 1997.

[78] G. Sun, J. Kurti, P. Rajczy, M. Kertesz, J. Hafner, and G. Kresse, J.

Mol. Struct.: THEOCHEM 624, 37 2003.

143

[79] S. B. Zhang, S. H. Wei, and Z. Zunger, Phys. Rev. B 63, 075205 2001.

[80] P. E. Blochl, Phys. Rev. B 50, 17953 1994.

[81] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 1993.

[82] B. Hammer, L. B. Hansen, and J. K. Nørskov, Phys. Rev. B 59, 7413

1999.

[83] E. L. Briggs, D. J. Sullivan, and J. Bernholc, Phys. Rev. B 54, 14362

1996.

[84] L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J. Chem.

Phys. 94, 7221 1991.

[85] J. Paier, R. Hirschl, M. Marsman, and G. Kresse, J. Chem. Phys. 122,

234102 2005.

[86] S. Goedecker, M. Teter and J. Hutter, Phys. Rev. B 54, 3 1996.

[87] S.G.Louie, S.Froyen and M.L.Cohen, Phys. Rev.B 26, 4 1982

[88] A. Zunger, Phys. Rev. B 22, 649 1980

[89] D. R. Hamann, M. Schluter, and C. Chiang, Phys.Rev. Lett. 43, 1494

1979

[90] M. T. Yin and M. L. Cohen, Phys. Rev. Lett. 45, 1004 1980

[91] R. Fournier, L. Boroukhovskaia, Theor. Chem. Acc. 112: 1–6 2004

[92] P. Rinke, C. G. V. de Walle, and M. Scheffler, Phys. Rev. Lett. 102,

026402 (2009).

[93] R. Nieminen, Modell. Simul. Mater. Sci. Eng. 17, 084001 (2009).

[94] A. F. Wright and N. A. Modine, Phys. Rev. B 74, 235209 (2006).

[95] J. R. Chelikowsky, N. Troullier, and Y. Saad, Phys. Rev. Lett. 72, 1240

1994[U+0352].

144

[96] E. L. Briggs, D. J. Sullivan, and J. Bernholc, Phys. Rev. B 52, R5471

1995.

[97] S. Ogut, H. Kim, J. R. Chelikowsky Phys. rev. B. (Rapid Comm.) 56,

18 1997.

[98] C.G. Van de Walle, J. Neugebauer, J. Appl. Phys. 95, 8 2004.

[99] R. S. Mulliken, J. Chem. Phys. 23, 1833 (1955).

[100] D.V. Makhov and L.J. Lewis, Phys. Rev. B 72, 073306 2005.

[101] S. Ogut and J. R. Chelikowsky Phys. Rev. Lett. 83, 19 (1999)

[102] M. Saito and A. Oshiyama, Phys. Rev. Lett. 73, 866 1994.

[103] P.Pulay, Chem. Phys. Lett. 73, 2, 393-398 (1990)

145

Acknowledgments

I am extremely grateful to Nanoscience Foundation who funded this project

for the first three years. All the staff members of the Foundation were very

helpful with the administrative proceedings like the VISA for me and my

wife, the reimbursements of the transport expenses, paper work for the tax

payment etc. I would like to thank them for such a support and wish them

all the best to carry on with the wonderful work of helping researchers from

all around the world.

The last four months of my PhD thesis was funded by CEA and I have

no words to thank for their kind consideration. I also thank all the adminis-

trative staff members to help me with all the necessary paper works needed

for such an extension.

To start with I would like to thank Dr. Damien Caliste, my PhD super-

visor, to have been so helpful. Through all the small and silly mistakes to

the biggest blunders he kept his support and helped me in all possible ways

to finish my work. There are so many important things that I learnt from

him. From computer techniques to concepts of physics discussions with him

always ended in my gaining some new knowledge. It was nice to know with

him the different aspects of research. I can’t thank him enough for his help

in difficult times of the research work or even the administrative hustles. I

thank him for his patience to deal with my innumerable mistakes in the re-

search work. It was very nice to know him and his family and I would be

ever thankful for his all round support.

I would take this opportunity to thank Dr. Thierry Deutsch, my director

to thesis, for letting me have the opportunity to work on this interesting

topic. I got to learn a lot about various aspects of physics and computer

science because of many important discussions with him. I am also thankful

for his kind consideration in letting me have an extension of my PhD con-

tract, which otherwise is very difficult to have as I gather. There had been

146

strained patches in between but I would take the positive out of it to thank

you for letting me know the hardships of research and I guess I have learnt

many important things to do a good research. Thanks a lot for being there.

I would like to thank Mr. Alex Willand and Dr. Stefan Goedecker for

collaborating on a project with me. Discussion through email with them had

been of great help in the research.

In my laboratory there are so many people to whom I was thankful for

various reasons. A special thanks to Dr. Luigi Genovese for some delightful

discussions regarding the collaborative work we did. It was great to learn so

many different aspects of physics and computer science from you. I would

like to thank Dr. Pascal Pochet, Dr. Yann-Michel Niquet, Dr. Frederic

Lancon and Dr. Brice Videau, discussions with whom helped me to learn a

lot about different things in physics and computer science. They had been

very understanding and patient to all my questions irrespective of their being

related to science or something else.

I had wonderful friends in the lab in the past three and half years. Each

of them helped me in all possible ways and words fail me when I wish to

thank them for being there with me. I would like to thank Thomas Jour-

dan, at whose party for the first time I got to know my friends very well. I

am thankful to Emanuel Arras, who helped me with my initial proceedings

in research with all sorts of things in physics and computers. I am equally

thankful to Martin Presson and Margaret Gabriel Presson for their wonderful

companionship. A very special thanks to Ivetta Slipukhina for being there as

a constant support encouraging to be positive in times when I was down. I

would like to thank Dulce Camacho for being such a wonderful friend, sharing

innumerable coffees and chats of all kinds. She has also been a great source of

inspiration. I would also thank Hector Mera for his tremendous support for

me and my work and for the wonderful times that we got to spend together.

His insight to science and inspiration to think differently have helped me a lot

in my journey. To my friend Cornelia Metzig I would be extremely grateful

147

as well for she helped me in many ways. Discussion with her about physics,

science in general, society, philosophy and many other things made my stay

in the lab very enjoyable. I would also thank Eduardo Machado Charry for

helping me with many small but important things regarding my thesis. His

presence always lights up the place. I would also like to thank Irinia Georgina

Groza for being a lovely inspiring and encouraging friend. My gratitude goes

for Bhaaraathi Natarajan, discussions with whom is something which I had

always enjoyed.

During my stay in France, there is a family to whom I will never be able

to say how grateful I am to have known them - Genvieve Stella Moriceau

and Hubert Moriceau. They have love us like their own and have helped us

in every possible way to make our lives safe, healthy and enjoyable. They

had been the greatest strength and I would cherish every moment of those

I spent with them. With them I got to know so much about the culture in

France and rest of Europe. They had been the haven where I would run every

time I am in some trouble and of course they would help me with whatever

they could. It’s been an honor to have known them so closely.

In these three and half years I became friends with so many people. Of

them all the one who became almost a part of me was Akash Chakraborty.

Now, to look back in the times that we spent together, I feel it was a part of

myself that was moving around with me. We drank and ate all that we could,

went to places together, watched and discussed about innumerable movies,

talked about all that we could find under the sun and most of all had the

best possible time between us. In days of hard work and distress his support

was like the protective imperceptible ether. Can I thank him at all for being

a part of my senses?

Friends like Soumen Mondal, Anita Sarkar, Dibyendu Hazra, Subhadeep

Dutta, Kalpana Mondal became more than family. Innumerable discussions

regarding various subjects that spanned our work, culture, personal life etc.

made our days in Grenoble so beautiful. A very special thanks to Soumen

148

Mondal for helping in with his ambidextrous power of mending household

things, giving opinions in research and entrepreneurship way of life. I would

also like to extend my gratitude to friends like Ayan Kumar and Kuheli Ban-

dopadhyay, Chandan Bera, Sutirtha and Dipanwita Mukherjee, Arijit and

Satarupa Roy, Abhijit Ghosh, Anupam Kundu, Sreeram Venkatpathy, Shilpi

Sreeram and Pragyan, Roopak Sinha, Arpita Sinha, Dhrovi and Diu for be-

ing there and making the days in Grenoble happy and enjoyable for us. I

would like to thank Biplab Mondal for lending me his camera for most of my

stay here in Grenoble. It has been really like a family I wish it stays like that

forever. Ananda Sankar Basu and Priyadarshini Chatterjee deserve a special

bunch of thanks for being almost always the driving force of this extended

family of ours. It’s been a privilege to know you all.

This thesis is probably an important milestone in this journey and I guess

it had started under the guidance of the teachers who I think I can never

properly thank. Dr. Ananda Dasgupta is one of those teachers to whom I

feel I could go at any point of time to ask any question regarding physics. It

is actually a great feeling of confidence to have when day in day out some-

one is struggling to reach an intellectual height. I would never be able to

thank him enough. Prof. Albert Gomes had been an infinite source of in-

spiration with his intense dedication for students and science. I would like

to express my gratitude for his being there with me all those times when I

needed inspiration to remain dedicated, focused and honest. I would like to

thank Dr. Dipankar Sengupta, Dr. Indranath Chaudhury, Dr. Tapati Dutta,

Dr. Shibaji Banerjee and Dr. Suman Bandopadhyay who had taught me so

many things in science and have always inspired. I am extremely thankful

to Prof. D.G. Kanhere who taught me so many different important things

in physics and computer science. This PhD wouldn’t have been possible if

it wasn’t him who inspired me to come here and work. He has been a great

source of inspiration to all the students as he has been for me. I would also

thank Dr. Anjali Kshirsagar, with whom I managed to learn various aspect

of computational physics and physics in general. I would also like to thank

Mr. Pinaki Mitra and Mr. Nimai Bhattacharya who always have preached

149

to me the reason to use life for the quest of higher knowledge of all kinds.

I am grateful to Mrs. Jayati Acharya whose good wishes and relentless en-

couragement along with her sublime music lessons helped me a lot to grow

as a human being and sustain under tremendous pressure.

Honestly it was my mother Mrs. Chhabi Deb who dreamt her son becoming a

scientist one day. The constant motivating factor above all was my mother’s

dream about this day. I cannot thank her enough for having had that dream

and pushing me hard enough so that I could reach where I am now. My

father Mr. Amit Krishna Deb, with his utmost honesty and sincerity to life

has forced me to follow the path of honesty and integrity even when I could

have drifted. It’s their sincere love and care for me which made me worthy

of whatever I have achieved till date. I am glad that I have a brother like

Mr. Ayan Krishna Deb, who loved me like his own self and even in the most

difficult of our days his love was as true as the sun. I would like to thank

Mr. Asesh Krishna Deb, my eldest cousin whose constant support always

provided me with tremendous confidence. I would also like to thank my all

other family members and all other friends whose love and affection had been

a constant source of inspiration.

I would like to take this opportunity to express my gratitude for the late

Mr. Samir Mukherjee who taught me through the first days of science and

mathematics. I would like to thank my oldest school friends Joy Banerjee

and Angshuman Samadder who had been with me though thick and thin and

who have inspired me through examples all these days.

Among the friends whose constant help saw me through the initial days

of the journey with physics, I would like to thank Arya Dhar, Amit Kumar

Pal, Sayan Chakraborty, Beas Roy, and Sumitra Raymoulik. They taught

me all that was in syllabus, made notes for me to pass, even introduced me

to various new subjects. It is great to know them so closely. I would like

to thank my friends Aditya Rotti, Jaydeep Belapure, Sumati Patil, Deep-

ashri Saraf, Narendranath Patra, Sayan Mondal, Anirban Pal and Saurav

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Roy because they always believed I could reach this far. I would take this

opportunity to thank our Ajji Mrs. Subha Joshi who have always loved me

so much and have inspired me immensely to thrive harder for my goal in life.

I would also thank my friend Maitra, whose constant adulation and indul-

gence made me believe that I can be a creative individual as well. Her

support, encouragement and friendship is something which one can always

take pride in. Her honesty and integrity to life itself have been a great source

of inspiration to me and my family.

In the end, I would like to thank the person, thanking whom seems ab-

solutely insignificant, for it is Mrs. Maitreyee Mookherjee, who essentially

lived a life completely for my cause and sake. That she gave up her own

career to be with me through the times I needed her more than ever, that

she had faith in me when even I had lost faith in life, that she loved me even

when I failed her, are reasons enough for not thanking her. One cannot thank

someone who has done so much. Her dedication and love for me has been

my energy so far. I couldn’t have made it anywhere without her constant

support, encouragement and love. I could have written pages for her but it

would mean the same - that I wish she stays there with me forever.

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